The Fredrickson-Andersen one spin facilitated model (FA-1f) on $\mathbb Z$ belongs to the class of kinetically constrained spin models (KCM). Each site refreshes with rate one its occupation variable to empty (respectively occupied) with probability $q$ (respectively $p=1-q$), provided at least one nearest neighbor is empty. Here, we study the non equilibrium dynamics of FA-1f started from a configuration entirely occupied on the left half-line and focus on the evolution of the front, namely the position of the leftmost zero. We prove, for $q$ larger than a threshold $\bar q<1$, a law of large numbers and a central limit theorem for the front, as well as the convergence to an invariant measure of the law of the process seen from the front.

We study heavy subtrees of conditional Galton-Watson trees. In a standard Galton-Watson tree conditional on its size being $n$, we order all children by their subtree sizes, from large (heavy) to small. A node is marked if it is among the $k$ heaviest nodes among its siblings. Unmarked nodes and their subtrees are removed, leaving only a tree of marked nodes, which we call the $k$-heavy tree. We study various properties of these trees, including their size and the maximal distance from any original node to the $k$-heavy tree. In particular, under some moment condition, the $2$-heavy tree is with high probability larger than $cn$ for some constant $c > 0$, and the maximal distance from the $k$-heavy tree is $O(n^{1/(k+1)})$ in probability. As a consequence, for uniformly random Apollonian networks of size $n$, the expected size of the longest simple path is $\Omega (n)$. We also show that the length of the heavy path (that is, $k=1$) converges (after rescaling) to the corresponding object in Aldous’ Brownian continuum random tree.

We close an unexpected gap in the literature of Stochastic Differential Equations (SDEs) with drifts of super linear growth and with random coefficients, namely, we prove Malliavin and Parametric Differentiability of such SDEs. The former is shown by proving Stochastic Gâteaux Differentiability and Ray Absolute Continuity. This method enables one to take limits in probability rather than mean square or almost surely bypassing the potentially non-integrable error terms from the unbounded drift. This issue is strongly linked with the difficulties of the standard methodology of [13, Lemma 1.2.3] for this setting. Several examples illustrating the range and scope of our results are presented.

We close with parametric differentiability and recover representations linking both derivatives as well as a Bismut-Elworthy-Li formula.

We establish an existence and uniqueness result for a class of multidimensional quadratic backward stochastic differential equations (BSDE). This class is characterized by constraints on some uniform a priori estimate on solutions of a sequence of approximated BSDEs. We also present effective examples of applications. Our approach relies on the strategy developed by Briand and Elie in [Stochastic Process. Appl. 123 2921–2939] concerning scalar quadratic BSDEs.

We consider sequences of symmetric $U$-statistics, not necessarily Hoeffding-degenerate, both in a one- and multi-dimensional setting, and prove quantitative central limit theorems (CLTs) based on the use of contraction operators. Our results represent an explicit counterpart to analogous criteria that are available for sequences of random variables living on the Gaussian, Poisson or Rademacher chaoses, and are perfectly tailored for geometric applications. As a demonstration of this fact, we develop explicit bounds for subgraph counting in generalised random graphs on Euclidean spaces; special attention is devoted to the so-called ‘dense parameter regime’ for uniformly distributed points, for which we deduce CLTs that are new even in their qualitative statement, and that substantially extend classical findings by Jammalamadaka and Janson (1986) and Bhattacharaya and Ghosh (1992).

We establish some deviation inequalities, moment bounds and almost sure results for the Wasserstein distance of order $p\in [1, \infty )$ between the empirical measure of independent and identically distributed ${\mathbb R}^d$-valued random variables and the common distribution of the variables. We only assume the existence of a (strong or weak) moment of order $rp$ for some $r>1$, and we discuss the optimality of the bounds.

A self-similar growth-fragmentation describes the evolution of particles that grow and split as time passes. Its genealogy yields a self-similar continuum tree endowed with an intrinsic measure. Extending results of Haas [13] for pure fragmentations, we relate the existence of an absolutely continuous profile to a simple condition in terms of the index of self-similarity and the so-called cumulant of the growth-fragmentation. When absolutely continuous, we approximate the profile by a function of the small fragments, and compute the Hausdorff dimension in the singular case. Applications to Boltzmann random planar maps are emphasized, exploiting recently established connections between growth-fragmentations and random maps [1, 5].

This paper studies the problem of equivalence of Gaussian measures induced by Gaussian random fields (GRFs) with stationary increments and proves a sufficient condition for the equivalence in terms of the behavior of the spectral measures at infinity. The main results extend those of Stein (2004), Van Zanten (2007, 2008) and are applicable to a rich family of nonstationary space-time models with possible anisotropy behavior.

We study the distribution of the least singular value associated to an ensemble of sparse random matrices. Our motivating example is the ensemble of $N\times N$ matrices whose entries are chosen independently from a Bernoulli distribution with parameter $p$. These matrices represent the adjacency matrices of random Erdős–Rényi digraphs and are sparse when $p\ll 1$. We prove that in the regime $pN\gg 1$, the distribution of the least singular value is universal in the sense that it is independent of $p$ and equal to the distribution of the least singular value of a Gaussian matrix ensemble. We also prove the universality of the joint distribution of multiple small singular values. Our methods extend to matrix ensembles whose entries are chosen from arbitrary distributions that may be correlated, complex valued, and have unequal variances.

For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}$, we consider the process of its eigenvalues $\{(\lambda _{1}^{(n)}(t),\dots , \lambda _{n}^{(n)}(t)); t\ge 0\}$ as well as its corresponding process of empirical spectral measures $\mu ^{(n)}=(\mu _{t}^{(n)}; t\geq 0)$. Under some mild conditions on the covariance function associated to $Y^{(n)}$, we prove that the process $\mu ^{(n)}$ converges in probability to a deterministic limit $\mu $, in the topology of uniform convergence over compact sets. We show that the process $\mu $ is characterized by its Cauchy transform, which is a rescaling of the solution of a Burgers’ equation. Our results extend those of Rogers and Shi [14] for the free Brownian motion and Pardo et al. [12] for the non-commutative fractional Brownian motion when $H>1/2$ whose arguments use strongly the non-collision of the eigenvalues. Our methodology does not require the latter property and in particular explains the remaining case of the non-commutative fractional Brownian motion for $H< 1/2$ which, up to our knowledge, was unknown.

We study the small-time asymptotics for hypoelliptic diffusion processes conditioned by their initial and final positions, in a model class of diffusions satisfying a weak Hörmander condition where the diffusivity is constant and the drift is linear. We show that, while the diffusion bridge can exhibit a blow-up behaviour in the small time limit, we can still make sense of suitably rescaled fluctuations which converge weakly. We explicitly describe the limit fluctuation process in terms of quantities associated to the unconditioned diffusion. In the discussion of examples, we also find an expression for the bridge from $0$ to $0$ in time $1$ of an iterated Kolmogorov diffusion.

In this paper, we are interested in deriving non-asymptotic error bounds for the multilevel Monte Carlo method. As a first step, we deal with the explicit Euler discretization of stochastic differential equations with a constant diffusion coefficient. We prove that, as long as the deviation is below an explicit threshold, a Gaussian-type concentration inequality optimal in terms of the variance holds for the multilevel estimator. To do so, we use the Clark-Ocone representation formula and derive bounds for the moment generating functions of the squared difference between a crude Euler scheme and a finer one and of the squared difference of their Malliavin derivatives.

The purpose of this note is to show how simple Optimal Transport arguments, on the real line, can be used in Superconcentration theory. This methodology is efficient to produce sharp non-asymptotic variance bounds for various functionals (maximum, median, $l^p$ norms) of standard Gaussian random vectors in $\mathbb{R} ^n$. The flexibility of this approach can also provide exponential deviation inequalities reflecting preceding variance bounds. As a further illustration, usual laws from Extreme theory and Coulomb gases are studied.

We study the stochastic wave equation with multiplicative noise and singular drift:

\[\partial _{t}u(t,x)=\Delta u(t,x)+u^{-\alpha }(t,x)+g(u(t,x))\dot{W} (t,x)\]

where $x$ lies in the circle $\mathbf{R} /J\mathbf{Z} $ and $u(0,x)>0$. We show that

(i) If $0<\alpha <1$ then with positive probability, $u(t,x)=0$ for some $(t,x)$.

(ii) If $\alpha >3$ then with probability one, $u(t,x)\ne 0$ for all $(t,x)$.

We consider a super-critical Galton-Watson tree $\tau $ whose non-degenerate offspring distribution has finite mean. We consider the random trees $\tau _n$ distributed as $\tau $ conditioned on the $n$-th generation, $Z_n$, to be of size $a_n\in{\mathbb N} $. We identify the possible local limits of $\tau _n$ as $n$ goes to infinity according to the growth rate of $a_n$. In the low regime, the local limit $\tau ^0$ is the Kesten tree, in the moderate regime the family of local limits, $\tau ^\theta $ for $\theta \in (0, +\infty )$, is distributed as $\tau $ conditionally on $\{W=\theta \}$, where $W$ is the (non-trivial) limit of the renormalization of $Z_n$. In the high regime, we prove the local convergence towards $\tau ^\infty $ in the Harris case (finite support of the offspring distribution) and we give a conjecture for the possible limit when the offspring distribution has some exponential moments. When the offspring distribution has a fat tail, the problem is open. The proof relies on the strong ratio theorem for Galton-Watson processes. Those latter results are new in the low regime and high regime, and they can be used to complete the description of the (space-time) Martin boundary of Galton-Watson processes. Eventually, we consider the continuity in distribution of the local limits $(\tau ^\theta , \theta \in [0, \infty ])$.

Consider the Totally Asymmetric Simple Exclusion Process (TASEP) on the integer lattice $ \mathbb{Z} $. We study the functional Large Deviations of the integrated current $ \mathsf{h} (t,x) $ under the hyperbolic scaling of space and time by $ N $, i.e., $ \mathsf{h} _{N}(t,\xi ) := \frac{1} {N}\mathsf{h} (Nt,N\xi ) $. As hinted by the asymmetry in the upper- and lower-tail large deviations of the exponential Last Passage Percolation, the TASEP exhibits two types of deviations. One type of deviations occur with probability $ \exp (-O(N)) $, referred to as speed-$ N $; while the other with probability $ \exp (-O(N^{2})) $, referred to as speed-$ N^2 $. In this work we study the speed-$ N^2 $ functional Large Deviation Principle (LDP) of the TASEP, and establish (non-matching) large deviation upper and lower bounds.

We investigate the upper tail probabilities of the all-time maximum of a stable Lévy process with a power negative drift. The asymptotic behaviour is shown to be exponential in the spectrally negative case and polynomial otherwise, with explicit exponents and constants. Analogous results are obtained, at a less precise level, for the fractionally integrated stable Lévy process. We also study the lower tail probabilities of the integrated stable Lévy process in the presence of a power positive drift.

A uniform attachment tree is a random tree that is generated dynamically. Starting from a fixed “seed” tree, vertices are added sequentially by attaching each vertex to an existing vertex chosen uniformly at random. Upon observing a large (unlabeled) tree, one wishes to find the initial seed. We investigate to what extent seed trees can be recovered, at least partially. We consider three types of seeds: a path, a star, and a random uniform attachment tree. We propose and analyze seed-finding algorithms for all three types of seed trees.

We propose a general method for investigating scaling limits of finite dimensional Markov chains to diffusions with jumps. The results of tightness, identification and convergence in law are based on the convergence of suitable characteristics of the chain transition. We apply these results to population processes recursively defined as sums of independent random variables. Two main applications are developed. First, we extend the Wright-Fisher model to independent and identically distributed random environments and show its convergence, under a large population assumption, to a Wright-Fisher diffusion in random environment. Second, we obtain the convergence in law of generalized Galton-Watson processes with interaction in random environment to solutions of stochastic differential equations with jumps.

The Brownian web (BW) is a collection of coalescing Brownian paths $(W_{(x,t)},(x,t) \in \mathbb{R} ^2)$ indexed by the plane. It appears in particular as continuous limit of various discrete models of directed forests of coalescing random walks and navigation schemes. Radial counterparts have been considered but global invariance principles are hard to establish. In this paper, we consider cylindrical forests which in some sense interpolate between the directed and radial forests: we keep the topology of the plane while still taking into account the angular component. We define in this way the cylindric Brownian web (CBW), which is locally similar to the planar BW but has several important macroscopic differences. For example, in the CBW, the coalescence time between two paths admits exponential moments and the CBW as its dual contain each a.s. a unique bi-infinite path. This pair of bi-infinite paths is distributed as a pair of reflected Brownian motions on the cylinder. Projecting the CBW on the radial plane, we obtain a radial Brownian web (RBW), i.e. a family of coalescing paths where under a natural parametrization, the angular coordinate of a trajectory is a Brownian motion. Recasting some of the discrete radial forests of the literature on the cylinder, we propose rescalings of these forests that converge to the CBW, and deduce the global convergence of the corresponding rescaled radial forests to the RBW. In particular, a modification of the radial model proposed in Coletti and Valencia is shown to converge to the CBW.

We consider the hydrodynamic limit in the macroscopic regime of the coupled system of stochastic differential equations,

\[ d\lambda _t^i=\frac{1} {\sqrt{N} } dW_t^i - V'(\lambda _t^i) dt+ \frac{\beta } {2N} \sum _{j\not =i} \frac{dt} {\lambda ^i_t-\lambda ^j_t}, \qquad i=1,\ldots ,N, \qquad \mbox{(0.1)} \]

with $\beta >1$, sometimes called generalized Dyson’s Brownian motion, describing the dissipative dynamics of a log-gas of $N$ equal charges with equilibrium measure corresponding to a $\beta $-ensemble, with sufficiently regular convex potential $V$. The limit $N\to \infty $ is known to satisfy a mean-field Mc Kean-Vlasov equation. Fluctuations around this limit have been shown [39] to define a Gaussian process solving some explicit martingale problem written in terms of a generalized transport equation.

We prove a series of results concerning either the Mc Kean-Vlasov equation for the density $\rho _t$, notably regularity results and time-evolution of the support, or the associated hydrodynamic fluctuation process, whose space-time covariance kernel we compute explicitly.

We consider the exclusion process on segments of the integers in a site-dependent random environment. We assume to be in the ballistic regime in which a single particle has positive linear speed. Our goal is to study the mixing time of the exclusion process when the number of particles is linear in the size of the segment. We investigate the order of the mixing time depending on the support of the environment distribution. In particular, we prove for nestling environments that the order of the mixing time is different than in the case of a single particle.

We consider biased random walks in a one-dimensional percolation model. This model goes back to Axelson-Fisk and Häggström and exhibits the same phase transition as biased random walk on the infinite cluster of supercritical Bernoulli bond percolation on $\mathbb{Z} ^d$, namely, for some critical value $\lambda _{\mathrm{c} }>0$ of the bias, it holds that the asymptotic linear speed $\overline{\mathrm {v}} $ of the walk is strictly positive if the bias $\lambda $ is strictly smaller than $\lambda _{\mathrm{c} }$, whereas $\overline{\mathrm {v}} =0$ if $\lambda \geq \lambda _{\mathrm{c} }$.

We show that at the critical bias $\lambda = \lambda _{\mathrm{c} }$, the displacement of the random walk from the origin is of order $n/\log n$. This is in accordance with simulation results by Dhar and Stauffer for biased random walk on the infinite cluster of supercritical bond percolation on $\mathbb{Z} ^d$.

Our result is based on fine estimates for the tails of suitable regeneration times. As a by-product of these estimates we also obtain the order of fluctuations of the walk in the sub-ballistic and in the ballistic, nondiffusive phase.

The concept of splitting tessellations and splitting tessellation processes in spherical spaces of dimension $d\geq 2$ is introduced. Expectations, variances and covariances of spherical curvature measures induced by a splitting tessellation are studied using tools from spherical integral geometry. Also the spherical pair-correlation function of the $(d-1)$-dimensional Hausdorff measure is computed explicitly and compared to its analogue for Poisson great hypersphere tessellations. Finally, the typical cell distribution and the distribution of the typical spherical maximal face of any dimension $k\in \{1,\ldots ,d-1\}$ are expressed as mixtures of the related distributions of Poisson great hypersphere tessellations. This in turn is used to determine the expected length and the precise birth time distribution of the typical maximal spherical segment of a splitting tessellation.

We correct a few more minor errors in our paper, Electron. J. Probab. 12, Paper 54 (2007), 1454–1508.

We investigate the problem of quantifying contraction coefficients of Markov transition kernels in Kantorovich ($L^1$ Wasserstein) distances. For diffusion processes, relatively precise quantitative bounds on contraction rates have recently been derived by combining appropriate couplings with carefully designed Kantorovich distances. In this paper, we partially carry over this approach from diffusions to Markov chains. We derive quantitative lower bounds on contraction rates for Markov chains on general state spaces that are powerful if the dynamics is dominated by small local moves. For Markov chains on $\mathbb R^d$ with isotropic transition kernels, the general bounds can be used efficiently together with a coupling that combines maximal and reflection coupling. The results are applied to Euler discretizations of stochastic differential equations with non-globally contractive drifts, and to the Metropolis adjusted Langevin algorithm for sampling from a class of probability measures on high dimensional state spaces that are not globally log-concave.

We establish two theorems for assessing the accuracy in total variation of multivariate discrete normal approximation to the distribution of an integer valued random vector $W$. The first is for sums of random vectors whose dependence structure is local. The second applies to random vectors $W$ resulting from integrating the ${\mathbb Z}^d$-valued marks of a marked point process with respect to its ground process. The error bounds are of magnitude comparable to those given in [Rinott & Rotar (1996)], but now with respect to the stronger total variation distance. Instead of requiring the summands to be bounded, we make third moment assumptions. We demonstrate the use of the theorems in four applications: monochrome edges in vertex coloured graphs, induced triangles and $2$-stars in random geometric graphs, the times spent in different states by an irreducible and aperiodic finite Markov chain, and the maximal points in different regions of a homogeneous Poisson point process.

Random population dynamics with catastrophes (events pertaining to possible elimination of a large portion of the population) has a long history in the mathematical literature. In this paper we study an ergodic model for random population dynamics with linear growth and binomial catastrophes: in a catastrophe, each individual survives with some fixed probability, independently of the rest. Through a coupling construction, we obtain sharp two-sided bounds for the rate of convergence to stationarity which are applied to show that the model exhibits a cutoff phenomenon.

We develop a multidimensional Stein methodology for non-degenerate self-decomposable random vectors in $\mathbb{R} ^d$ having finite first moment. Building on previous univariate findings, we solve an integro-partial differential Stein equation by a mixture of semigroup and Fourier analytic methods. Then, under a second moment assumption, we introduce a notion of Stein kernel and an associated Stein discrepancy specifically designed for infinitely divisible distributions. Combining these new tools, we obtain quantitative bounds on smooth-Wasserstein distances between a probability measure in $\mathbb{R} ^d$ and a non-degenerate self-decomposable target law with finite second moment. Finally, under an appropriate Poincaré-type inequality assumption, we investigate, via variational methods, the existence of Stein kernels. In particular, this leads to quantitative versions of classical results on characterizations of probability distributions by variational functionals.

We introduce a class of probability measure-valued diffusions, coined polynomial, of which the well-known Fleming–Viot process is a particular example. The defining property of finite dimensional polynomial processes considered in [8, 21] is transferred to this infinite dimensional setting. This leads to a representation of conditional marginal moments via a finite dimensional linear PDE, whose spatial dimension corresponds to the degree of the moment. As a result, the tractability of finite dimensional polynomial processes are preserved in this setting. We also obtain a representation of the corresponding extended generators, and prove well-posedness of the associated martingale problems. In particular, uniqueness is obtained from the duality relationship with the PDEs mentioned above.

We consider invasion percolation on Galton-Watson trees. On almost every Galton-Watson tree, the invasion cluster almost surely contains only one infinite path. This means that for almost every Galton-Watson tree, invasion percolation induces a probability measure on infinite paths from the root. We show that under certain conditions of the progeny distribution, this measure is absolutely continuous with respect to the limit uniform measure. This confirms that invasion percolation, an efficient self-tuning algorithm, may be used to sample approximately from the limit uniform distribution. Additionally, we analyze the forward maximal weights along the backbone of the invasion cluster and prove a limit law for the process.

Given a graph $G=(V,E)$, consider Poisson($|V|$) walkers performing independent lazy simple random walks on $G$ simultaneously, where the initial position of each walker is chosen independently with probability proportional to the degrees. When two walkers visit the same vertex at the same time they are declared to be acquainted. The social connectivity time $\mathrm{SC} (G)$ is defined as the first time in which there is a path of acquaintances between every pair of walkers. It is shown that when the average degree of $G$ is $d$, with high probability \[ c\log |V| \le \mathrm{SC} (G) \le C d^{1+5 \cdot 1_{G \text{ is not regular} } } \log ^3 |V|. \] When $G$ is regular the lower bound is improved to $\mathrm{SC} (G) \ge \log |V| -6 \log \log |V| $, with high probability. We determine $\mathrm{SC} (G)$ up to a constant factor in the cases that $G$ is an expander and when it is the $n$-cycle.

The boundary behavior of continuous-state branching processes with quadratic competition is studied in whole generality. We first observe that despite competition, explosion can occur for certain branching mechanisms. We obtain a necessary and sufficient condition for $\infty $ to be accessible in terms of the branching mechanism and the competition parameter $c>0$. We show that when $\infty $ is inaccessible, it is always an entrance boundary. In the case where $\infty $ is accessible, explosion can occur either by a single jump to $\infty $ (the process at $z$ jumps to $\infty $ at rate $\lambda z$ for some $\lambda >0$) or by accumulation of large jumps over finite intervals. We construct a natural extension of the minimal process and show that when $\infty $ is accessible and $0\leq \frac{2\lambda } {c}<1$, the extended process is reflected at $\infty $. In the case $\frac{2\lambda } {c}\geq 1$, $\infty $ is an exit of the extended process. When the branching mechanism is not the Laplace exponent of a subordinator, we show that the process with reflection at $\infty $ gets extinct almost surely. Moreover absorption at $0$ is almost sure if and only if Grey’s condition is satisfied. When the branching mechanism is the Laplace exponent of a subordinator, necessary and sufficient conditions are given for a stationary distribution to exist. The Laplace transform of the latter is provided. The study is based on classical time-change arguments and on a new duality method relating logistic CSBPs with certain generalized Feller diffusions.

Consider a random polynomial $Q_n$ of degree $n+1$ whose zeroes are i.i.d. random variables $\xi _0,\xi _1,\ldots ,\xi _n$ in the complex plane. We study the pairing between the zeroes of $Q_n$ and its critical points, i.e. the zeroes of its derivative $Q_n'$. In the asymptotic regime when $n\to \infty $, with high probability there is a critical point of $Q_n$ which is very close to $\xi _0$. We localize the position of this critical point by proving that the difference between $\xi _0$ and the critical point has approximately complex Gaussian distribution with mean $1/(nf(\xi _0))$ and variance of order $\log n \cdot n^{-3}$. Here, $f(z)= \mathbb E [\frac 1 {z-\xi _k}]$ is the Cauchy–Stieltjes transform of the $\xi _k$’s. We also state some conjectures on critical points of polynomials with dependent zeroes, for example the Weyl polynomials and characteristic polynomials of random matrices.

We obtain a general bound for the Wasserstein-2 distance in normal approximation for sums of locally dependent random variables. The proof is based on an asymptotic expansion for expectations of second-order differentiable functions of the sum. We apply the main result to obtain Wasserstein-2 bounds in normal approximation for sums of $m$-dependent random variables, U-statistics and subgraph counts in the Erdős-Rényi random graph. We state a conjecture on Wasserstein-$p$ bounds for any positive integer $p$ and provide supporting arguments for the conjecture.

We study some linear SDEs arising from the two-dimensional $q$-Whittaker driven particle system on the torus as $q\to 1$. The main result proves that the SDEs along certain characteristics converge to the additive stochastic heat equation. Extensions for the SDEs with generalized coefficients and in other spatial dimensions are also obtained. Our proof views the limiting process after recentering as a process of the convolution of a space-time white noise and the Fourier transform of the heat kernel. Accordingly we turn to similar space-time stochastic integrals defined by the SDEs, but now the convolution and the Fourier transform are broken. To obtain tightness of these induced integrals, we bound the oscillations of complex exponentials arising from divergence of the characteristics, with two methods of different nature.

We consider confinement properties of families of non-colliding Brownian bridges above a hard wall, which are subject to geometrically growing self-potentials of tilted area type. The model is introduced in order to mimic level lines of $2+1$ discrete Solid-On-Solid random interfaces above a hard wall.

In previous work by Avena and den Hollander [3], a model of a random walk in a dynamic random environment was proposed where the random environment is resampled from a given law along a given sequence of times. In the regime where the increments of the resampling times diverge, which is referred to as the cooling regime, a weak law of large numbers and certain fluctuation properties were derived under the annealed measure, in dimension one. In the present paper we show that a strong law of large numbers and a quenched large deviation principle hold as well. In the cooling regime, the random walk can be represented as a sum of independent variables, distributed as the increments of a random walk in a static random environment over diverging periods of time. Our proofs require suitable multi-layer decompositions of sums of random variables controlled by moment bounds and concentration estimates. Along the way we derive two results of independent interest, namely, concentration inequalities for the random walk in the static random environment and an ergodic theorem that deals with limits of sums of triangular arrays representing the structure of the cooling regime. We close by discussing our present understanding of homogenisation effects as a function of the cooling scheme, and by hinting at what can be done in higher dimensions. We argue that, while the cooling scheme does not affect the speed in the strong law of large numbers nor the rate function in the large deviation principle, it does affect the fluctuation properties.

We prove annealed scaling relations for planar Voronoi percolation. To our knowledge, this is the first result of this kind for a continuum percolation model. We are mostly inspired by the proof of scaling relations for Bernoulli percolation by Kesten [22]. Along the way, we show an annealed quasi-multiplicativity property by relying on the quenched box-crossing property proved by Ahlberg, Griffiths, Morris and Tassion [3]. Intermediate results also include the study of quenched and annealed notions of pivotal events and the extension of the quenched box-crossing property of [3] to the near-critical regime.

We study the Wiener–Hopf factorization for Lévy processes $X_{t}$ with completely monotone jumps. Extending previous results of L.C.G. Rogers, we prove that the space-time Wiener–Hopf factors are complete Bernstein functions of both the spatial and the temporal variable. As a corollary, we prove complete monotonicity of: (a) the tail of the distribution function of the supremum of $X_{t}$ up to an independent exponential time; (b) the Laplace transform of the supremum of $X_{t}$ up to a fixed time $T$, as a function of $T$. The proof involves a detailed analysis of the holomorphic extension of the characteristic exponent $f(\xi )$ of $X_{t}$, including a peculiar structure of the curve along which $f(\xi )$ takes real values.

Cellular automata (CA) are dynamical systems on symbolic configurations on the lattice. They are also used as models of massively parallel computers. As dynamical systems, one would like to understand the effect of small random perturbations on the dynamics of CA. As models of computation, they can be used to study the reliability of computation against noise.

We consider various families of CA (nilpotent, permutive, gliders, CA with a spreading symbol, surjective, algebraic) and prove that they are highly unstable against noise, meaning that they forget their initial conditions under slightest positive noise. This is manifested as the ergodicity of the resulting probabilistic CA. The proofs involve a collection of different techniques (couplings, entropy, Fourier analysis), depending on the dynamical properties of the underlying deterministic CA and the type of noise.

We present precise multilevel exponential concentration inequalities for polynomials in Ising models satisfying the Dobrushin condition. The estimates have the same form as two-sided tail estimates for polynomials in Gaussian variables due to Latała. In particular, for quadratic forms we obtain a Hanson–Wright type inequality.

We also prove concentration results for convex functions and estimates for nonnegative definite quadratic forms, analogous as for quadratic forms in i.i.d. Rademacher variables, for more general random vectors satisfying the approximate tensorization property for entropy.

Let $G$ be a compact Lie group, $N\geq 1$ and $L>0$. The random geometric graph on $G$ is the random graph $\Gamma _{\mathrm{geom} }(N,L)$ whose vertices are $N$ random points $g_1,\ldots ,g_N$ chosen under the Haar measure of $G$, and whose edges are the pairs $\{g_i,g_j\}$ with $d(g_i,g_j)\leq L$, $d$ being the distance associated to the standard Riemannian structure on $G$. In this paper, we describe the asymptotic behavior of the spectrum of the adjacency matrix of $\Gamma _{\mathrm{geom} }(N,L)$, when $N$ goes to infinity.

1. If $L$ is fixed and $N \to + \infty $ (Gaussian regime), then the largest eigenvalues of $\Gamma _{\mathrm{geom} }(N,L)$ converge after an appropriate renormalisation towards certain explicit linear combinations of values of Bessel functions.

2. If $L = O(N^{-\frac{1} {\dim G}})$ and $N \to +\infty $ (Poissonian regime), then the geometric graph $\Gamma _{\mathrm{geom} }(N,L)$ converges in the local Benjamini–Schramm sense, which implies the weak convergence in probability of the spectral measure of $\Gamma _{\mathrm{geom} }(N,L)$.

In both situations, the representation theory of the group $G$ provides us with informations on the limit of the spectrum, and conversely, the computation of this limiting spectrum involves many classical tools from representation theory: Weyl’s character formula and the weight lattice in the Gaussian regime, and a degeneration of these objects in the Poissonian regime. The representation theoretic approach allows one to understand precisely how the degeneration from the Gaussian to the Poissonian regime occurs, and the article is written so as to highlight this degeneration phenomenon. In the Poissonian regime, this approach leads us to an algebraic conjecture on certain functionals of the irreducible representations of $G$.

We consider a Gaussian random matrix with correlated entries that have a power law decay of order $d>2$ and prove universality for the extreme eigenvalues. A local law is proved using the self-consistent equation combined with a decomposition of the matrix. This local law along with concentration of eigenvalues around the edge allows us to get a bound for extreme eigenvalues. Using a recent result of the Dyson-Brownian motion, we prove universality of extreme eigenvalues.

The stochastic models investigated in this paper describe the evolution of a set of $F_N$ identical balls scattered into $N$ urns connected by an underlying symmetrical graph with constant degree $h_N$. After some random amount of time all the balls of any urn are redistributed locally, among the $h_N$ urns of its neighborhood. The allocation of balls is done at random according to a set of weights which depend on the state of the system. The main original features of this context is that the cardinality $h_N$ of the range of interaction is not necessarily linear with respect to $N$ as in a classical mean-field context and, also, that the number of simultaneous jumps of the process is not bounded due to the redistribution of all balls of an urn at the same time. The approach relies on the analysis of the evolution of the local empirical distributions associated to the state of urns located in the neighborhood of a given urn. Under convenient conditions, by taking an appropriate Wasserstein distance and by establishing several technical estimates for local empirical distributions, we are able to prove mean-field convergence results.

When the load per node goes to infinity, a convergence result for the invariant distribution of the associated McKean-Vlasov process is obtained for several allocation policies. For the class of power of $d$ choices policies, we show that the associated invariant measure has an asymptotic finite support property under this regime. This result differs somewhat from the classical double exponential decay property usually encountered in the literature for power of $d$ choices policies.

We study a random walk $\mathbf{S} _n$ on $\mathbb{Z} ^d$ ($d\geq 1$), in the domain of attraction of an operator-stable distribution with index $\boldsymbol{\alpha } =(\alpha _1,\ldots ,\alpha _d) \in (0,2]^d$: in particular, we allow the scalings to be different along the different coordinates. We prove a strong renewal theorem, i.e. a sharp asymptotic of the Green function $G(\mathbf{0} ,\mathbf{x} )$ as $\|\mathbf{x} \|\to +\infty $, along the “favorite direction or scaling”: (i) if $\sum _{i=1}^d \alpha _i^{-1} < 2$ (reminiscent of Garsia-Lamperti’s condition when $d=1$ [17]); (ii) if a certain local condition holds (reminiscent of Doney’s [13, Eq. (1.9)] when $d=1$). We also provide uniform bounds on the Green function $G(\mathbf{0} ,\mathbf{x} )$, sharpening estimates when $\mathbf{x} $ is away from this favorite direction or scaling. These results improve significantly the existing literature, which was mostly concerned with the case $\alpha _i\equiv \alpha $, in the favorite scaling, and has even left aside the case $\alpha \in [1,2)$ with non-zero mean. Most of our estimates rely on new general (multivariate) local large deviations results, that were missing in the literature and that are of interest on their own.

Crump–Mode–Jagers (CMJ) trees generalize Galton–Watson trees by allowing individuals to live for an arbitrary duration and give birth at arbitrary times during their life-time. In this paper, we exhibit a simple condition under which the height and contour processes of CMJ forests belong to the universality class of Bellman–Harris processes. This condition formalizes an asymptotic independence between the chronological and genealogical structures. We show that it is satisfied by a large class of CMJ processes and in particular, quite surprisingly, by CMJ processes with a finite variance offspring distribution. Along the way, we prove a general tightness result.

We prove an invariance principle for a class of zero-drift spatially non-homogeneous random walks in ${\mathbb R}^d$, which may be recurrent in any dimension. The limit $\mathcal{X} $ is an elliptic martingale diffusion, which may be point-recurrent at the origin for any $d\geq 2$. To characterize $\mathcal{X} $, we introduce a (non-Euclidean) Riemannian metric on the unit sphere in ${\mathbb R}^d$ and use it to express a related spherical diffusion as a Brownian motion with drift. This representation allows us to establish the skew-product decomposition of the excursions of $\mathcal{X} $ and thus develop the excursion theory of $\mathcal{X} $ without appealing to the strong Markov property. This leads to the uniqueness in law of the stochastic differential equation for $\mathcal{X} $ in ${\mathbb R}^d$, whose coefficients are discontinuous at the origin. Using the Riemannian metric we can also detect whether the angular component of the excursions of $\mathcal{X} $ is time-reversible. If so, the excursions of $\mathcal{X} $ in ${\mathbb R}^d$ generalize the classical Pitman–Yor splitting-at-the-maximum property of Bessel excursions.

For $d \geq 2$ and $n \in \mathbb{N} $, let $\mathsf{W} _n$ denote the uniform law on self-avoiding walks beginning at the origin in the integer lattice $\mathbb{Z} ^d$, and write $\Gamma $ for a $\mathsf{W} _n$-distributed walk. We show that the closing probability $\mathsf{W} _n \big (\vert \vert \Gamma _n \vert \vert = 1 \big )$ that $\Gamma $’s endpoint neighbours the origin is at most $n^{-4/7 + o(1)}$ for a positive density set of odd $n$ in dimension $d = 2$. This result is proved using the snake method, a general technique for proving closing probability upper bounds, which originated in [3] and was made explicit in [8]. Our conclusion is reached by applying the snake method in unison with a polygon joining technique whose use was initiated by Madras in [13].

We consider the free energy $F(\beta )$ of the directed polymers in random environment in $1+1$-dimension. It is known that $F(\beta )$ is of order $-\beta ^4$ as $\beta \to 0$ [3, 28, 42]. In this paper, we will prove that under a certain dimension free concentration condition on the potential, \[ \lim _{\beta \to 0}\frac{F(\beta )} {\beta ^4}=\lim _{T\to \infty }\frac{1} {T}P_\mathcal{Z} \left [\log \mathcal{Z} _{\sqrt{2} }(T)\right ] =-\frac{1} {6}, \] where $\{\mathcal{Z} _\beta (t,x):t\geq 0,x\in \mathbb{R} \}$ is the unique mild solution to the stochastic heat equation \[ \frac{\partial } {\partial t}\mathcal{Z} =\frac{1} {2}\Delta \mathcal{Z} +\beta \mathcal{Z} {\dot{\mathcal W} },\ \ \lim _{t\to 0}\mathcal{Z} (t,x)dx=\delta _{0}(dx), \] where $\mathcal{W} $ is a time-space white noise and \[ \mathcal{Z} _\beta (t)=\int _\mathbb{R} \mathcal{Z} _\beta (t,x)dx. \]

Mean field games (MFGs) describe the limit, as $n$ tends to infinity, of stochastic differential games with $n$ players interacting with one another through their common empirical distribution. Under suitable smoothness assumptions that guarantee uniqueness of the MFG equilibrium, a form of law of large of numbers (LLN), also known as propagation of chaos, has been established to show that the MFG equilibrium arises as the limit of the sequence of empirical measures of the $n$-player game Nash equilibria, including the case when player dynamics are driven by both idiosyncratic and common sources of noise. The proof of convergence relies on the so-called master equation for the value function of the MFG, a partial differential equation on the space of probability measures. In this work, under additional assumptions, we establish a functional central limit theorem (CLT) that characterizes the limiting fluctuations around the LLN limit as the unique solution of a linear stochastic PDE. The key idea is to use the solution to the master equation to construct an associated McKean-Vlasov interacting $n$-particle system that is sufficiently close to the Nash equilibrium dynamics of the $n$-player game for large $n$. We then derive the CLT for the latter from the CLT for the former. Along the way, we obtain a new multidimensional CLT for McKean-Vlasov systems. We also illustrate the broader applicability of our methodology by applying it to establish a CLT for a specific linear-quadratic example that does not satisfy our main assumptions, and we explicitly solve the resulting stochastic PDE in this case.

The goal of this work is to decompose random populations with a genealogy in subfamilies of a given degree of kinship and to obtain a notion of infinitely divisible genealogies. We model the genealogical structure of a population by (equivalence classes of) ultrametric measure spaces (um-spaces) as elements of the Polish space $\mathbb{U} $ which we recall. In order to then analyze the family structure in this coding we introduce an algebraic structure on um-spaces (a consistent collection of semigroups). This allows us to obtain a path of decompositions of subfamilies of fixed kinship $ h $ (described as ultrametric measure spaces), for every depth $ h $ as a measurable functional of the genealogy.

Technically the elements of the semigroup are those um-spaces which have diameter less or equal to $2h$ called $h$-forests ($h> 0$). They arise from a given ultrametric measure space by applying maps called $ h-$truncation. We can define a concatenation of two $ h$-forests as binary operation. The corresponding semigroup is a Delphic semigroup and any $h$-forest has a unique prime factorization in $h$-trees (um-spaces of diameter less than $2h$). Therefore we have a nested $\mathbb{R} ^{+}$-indexed consistent (they arise successively by truncation) collection of Delphic semigroups with unique prime factorization.

Random elements in the semigroup are studied, in particular infinitely divisible random variables. Here we define infinite divisibility of random genealogies as the property that the $h$-tops can be represented as concatenation of independent identically distributed h-forests for every $h$ and obtain a Lévy-Khintchine representation of this object and a corresponding representation via a concatenation of points of a Poisson point process of h-forests.

Finally the case of discrete and marked um-spaces is treated allowing to apply the results to both the individual based and most important spatial populations.

The results have various applications. In particular the case of the genealogical ($\mathbb{U} $-valued) Feller diffusion and genealogical ($\mathbb{U} ^{V}$-valued) super random walk is treated based on the present work in [13] and [24].

In the part II of this paper we go in a different direction and refine the study in the case of continuum branching populations, give a refined analysis of the Laplace functional and give a representation in terms of a Cox process on h-trees, rather than forests.

We define the (random) $k$-cut number of a rooted graph to model the difficulty of the destruction of a resilient network. The process is as the cut model of Meir and Moon [21] except now a node must be cut $k$ times before it is destroyed. The first order terms of the expectation and variance of $\mathcal{X} _{n}$, the $k$-cut number of a path of length $n$, are proved. We also show that $\mathcal{X} _{n}$, after rescaling, converges in distribution to a limit $\mathcal{B} _{k}$, which has a complicated representation. The paper then briefly discusses the $k$-cut number of some trees and general graphs. We conclude by some analytic results which may be of interest.

For a one-dimensional super-Brownian motion with density $X(t,x)$, we construct a random measure $L_{t}$ called the boundary local time which is supported on $BZ_{t} := \partial \{x:X(t,x) = 0\}$, thus confirming a conjecture of Mueller, Mytnik and Perkins [13]. $L_{t}$ is analogous to the local time at $0$ of solutions to an SDE. We establish first and second moment formulas for $L_{t}$, some basic properties, and a representation in terms of a cluster decomposition. Via the moment measures and the energy method we give a more direct proof that $\text{dim} (BZ_{t}) = 2-2\lambda _{0}> 0$ with positive probability, a recent result of Mueller, Mytnik and Perkins [13], where $-\lambda _{0}$ is the lead eigenvalue of a killed Ornstein-Uhlenbeck operator that characterizes the left tail of $X(t,x)$. In a companion work [6], the author and Perkins use the boundary local time and some of its properties proved here to show that $\text{dim} (BZ_{t}) = 2-2\lambda _{0}$ a.s. on $\{X_{t}(\mathbb{R} ) > 0 \}$.

Conditioning Markov processes to avoid a set is a classical problem that has been studied in many settings. In the present article we study the question if a Lévy process can be conditioned to avoid an interval and, if so, the path behavior of the conditioned process. For Lévy processes with finite second moments we show that conditioning is possible and identify the conditioned process as an $h$-transform of the original killed process. The $h$-transform is explicit in terms of successive overshoot distributions and is used to prove that the conditioned process diverges to $+\infty $ and $-\infty $ with positive probabilities.

Consider a heavy-tailed branching process (denoted by $Z_{n}$) in random environments, under the condition which infers that $\mathbb{E} \log m(\xi _{0})=\infty $. We show that (1) there exists no proper $c_{n}$ such that $\{Z_{n}/c_{n}\}$ has a proper, non-degenerate limit; (2) normalized by a sequence of functions, a proper limit can be obtained, i.e., $y_{n}\left (\bar{\xi } ,Z_{n}(\bar{\xi } )\right )$ converges almost surely to a random variable $Y(\bar{\xi } )$, where $Y\in (0,1)~\eta $-a.s.; (3) finally, we give the necessary and sufficient conditions for the almost sure convergence of $\left \{\frac{U(\bar {\xi },Z_{n}(\bar {\xi }))} {c_{n}(\bar{\xi } )}\right \}$, where $U(\bar{\xi } )$ is a slowly varying function that may depend on $\bar{\xi } $.

We study a one-dimensional spatial population model where the population sizes of the subpopulations are chosen according to a translation invariant and ergodic distribution and are uniformly bounded away from 0 and infinity. We suppose that the frequencies of a particular genetic type in the colonies evolve according to a system of interacting diffusions, following the stepping stone model of Kimura. We show that, over large spatial and temporal scales, this model behaves like the solution to a stochastic heat equation with Wright-Fisher noise with constant coefficients. These coefficients are the effective diffusion rate of genes within the population and the effective local population density. We find that, in our model, the local heterogeneity leads to a slower effective diffusion rate and a larger effective population density than in a uniform population. Our proof relies on duality techniques, an invariance principle for reversible random walks in a random environment and a convergence result for a system of coalescing random walks in a random environment.

A mated-CRT map is a random planar map obtained as a discretized mating of correlated continuum random trees. Mated-CRT maps provide a coarse-grained approximation of many other natural random planar map models (e.g., uniform triangulations and spanning tree-weighted maps), and are closely related to $\gamma $-Liouville quantum gravity (LQG) for $\gamma \in (0,2)$ if we take the correlation to be $-\cos (\pi \gamma ^{2}/4)$. We prove estimates for the Dirichlet energy and the modulus of continuity of a large class of discrete harmonic functions on mated-CRT maps, which provide a general toolbox for the study of the quantitative properties of random walk and discrete conformal embeddings for these maps.

For example, our results give an independent proof that the simple random walk on the mated-CRT map is recurrent, and a polynomial upper bound for the maximum length of the edges of the mated-CRT map under a version of the Tutte embedding. Our results are also used in other work by the first two authors which shows that for a class of random planar maps — including mated-CRT maps and the UIPT — the spectral dimension is two (i.e., the return probability of the simple random walk to its starting point after $n$ steps is $n^{-1+o_{n}(1)}$) and the typical exit time of the walk from a graph-distance ball is bounded below by the volume of the ball, up to a polylogarithmic factor.

We introduce a new framework to deal with rough differential equations based on flows and their approximations. Our main result is to prove that measurable flows exist under weak conditions, even if solutions to the corresponding rough differential equations are not unique. We show that under additional conditions of the approximation, there exists a unique Lipschitz flow. Then, a perturbation formula is given. Finally, we link our approach to the additive, multiplicative sewing lemmas and the rough Euler scheme.

We study the notions of mild solution and generalized solution to a linear stochastic partial differential equation driven by a pure jump symmetric Lévy white noise, with symmetric $\alpha $-stable Lévy white noise as an important special case. We identify conditions for existence of these two kinds of solutions, and, together with a new stochastic Fubini theorem, we provide conditions under which they are essentially equivalent. We apply these results to the linear stochastic heat, wave and Poisson equations driven by a symmetric $\alpha $-stable Lévy white noise.

We study the convergence of the population dynamics algorithm, which produces sample pools of random variables having a distribution that closely approximates that of the special endogenous solution to a variety of branching stochastic fixed-point equations, including the smoothing transform, the high-order Lindley equation, the discounted tree-sum and the free-entropy equation. Specifically, we show its convergence in the Wasserstein metric of order $p$ ($p \geq 1$) and prove the consistency of estimators based on the sample pool produced by the algorithm.

We study the recurrence/transience phase transition for Markov chains on ${\mathbb{R} }_{+}$, $\mathbb{R} $, and ${\mathbb{R} }^{2}$ whose increments have heavy tails with exponent in $(1,2)$ and asymptotically zero mean. This is the infinite-variance analogue of the classical Lamperti problem. On ${\mathbb{R} }_{+}$, for example, we show that if the tail of the positive increments is about $c y^{-\alpha }$ for an exponent $\alpha \in (1,2)$ and if the drift at $x$ is about $b x^{-\gamma }$, then the critical regime has $\gamma = \alpha -1$ and recurrence/transience is determined by the sign of $b + c\pi \operatorname{cosec} (\pi \alpha )$. On $\mathbb{R} $ we classify whether transience is directional or oscillatory, and extend an example of Rogozin & Foss to a class of transient martingales which oscillate between $\pm \infty $. In addition to our recurrence/transience results, we also give sharp results on the existence/non-existence of moments of passage times.

We study a system of branching Brownian motions on $\mathbb{R} $ with annihilation: at each branching time a new particle is created and the leftmost one is deleted. The case of strictly local creations (the new particle is put exactly at the same position of the branching particle) was studied in [10]. In [11] instead the position $y$ of the new particle has a distribution $p(x,y)dy$, $x$ the position of the branching particle, however particles in between branching times do not move. In this paper we consider Brownian motions as in [10] and non local branching as in [11] and prove convergence in the continuum limit (when the number $N$ of particles diverges) to a limit density which satisfies a free boundary problem when this has classical solutions. We use in the convergence a stronger topology than in [10] and [11] and have explicit bounds on the rate of convergence.

The interchange process on a finite graph is obtained by placing a particle on each vertex of the graph, then at rate $1$, selecting an edge uniformly at random and swapping the two particles at either end of this edge. In this paper we develop new techniques to show the existence of a phase transition of the interchange process on the $2$-dimensional Hamming graph. We show that in the subcritical phase, all of the cycles of the process have length $O(\log n)$, whereas in the supercritical phase a positive density of vertices lies in cycles of length at least $n^{2-\varepsilon }$ for any $\varepsilon >0$.

Recently, Holmes and Perkins identified conditions which ensure that for a class of critical lattice models the scaling limit of the range is the range of super-Brownian motion. One of their conditions is an estimate on a spatial moment of order higher than four, which they verified for the sixth moment for spread-out lattice trees in dimensions $d>8$. Chen and Sakai have proved the required moment estimate for spread-out critical oriented percolation in dimensions $d+1>4+1$. We use the lace expansion to prove estimates on all moments for the spread-out critical contact process in dimensions $d>4$, which in particular fulfills the spatial moment condition of Holmes and Perkins. Our method of proof is relatively simple, and, as we show, it applies also to oriented percolation and lattice trees. Via the convergence results of Holmes and Perkins, the upper bounds on the spatial moments can in fact be promoted to asymptotic formulas with explicit constants.

We consider the problem of optimally stopping a one-dimensional regular continuous strong Markov process with a stopping time satisfying an expectation constraint. We show that it is sufficient to consider only stopping times such that the law of the process at the stopping time is a weighted sum of 3 Dirac measures. The proof uses recent results on Skorokhod embeddings in order to reduce the stopping problem to a linear optimization problem over a convex set of probability measures.

We substantially refine asymptotic logarithmic upper bounds produced by Svante Janson (2015) on the right tail of the limiting QuickSort distribution function $F$ and by Fill and Hung (2018) on the right tails of the corresponding density $f$ and of the absolute derivatives of $f$ of each order. For example, we establish an upper bound on $\log [1 - F(x)]$ that matches conjectured asymptotics of Knessl and Szpankowski (1999) through terms of order $(\log x)^{2}$; the corresponding order for the Janson (2015) bound is the lead order, $x \log x$.

Using the refined asymptotic bounds on $F$, we derive right-tail large deviation (LD) results for the distribution of the number of comparisons required by QuickSort that substantially sharpen the two-sided LD results of McDiarmid and Hayward (1996).

We prove a shape theorem for Poisson cylinders, and give a power-law bound on surface fluctuations. In particular, we show that for any $a \in (1/2, 1)$, conditioned on the origin being in the set of cylinders, if a point belongs to this set and has Euclidean norm below $R$, then this point lies at internal distance less than $R + O(R^{a})$ from the origin.

A random walk in a sparse random environment is a model introduced by Matzavinos et al. [Electron. J. Probab. 21, paper no. 72: 2016] as a generalization of both a simple symmetric random walk and a classical random walk in a random environment. A random walk $(X_{n})_{n\in \mathbb{N} \cup \{0\}}$ in a sparse random environment $(S_{k},\lambda _{k})_{k\in \mathbb{Z} }$ is a nearest neighbor random walk on $\mathbb{Z} $ that jumps to the left or to the right with probability $1/2$ from every point of $\mathbb{Z} \setminus \{\ldots ,S_{-1},S_{0}=0,S_{1},\ldots \}$ and jumps to the right (left) with the random probability $\lambda _{k+1}$ ($1-\lambda _{k+1}$) from the point $S_{k}$, $k\in \mathbb{Z} $. Assuming that $(S_{k}-S_{k-1},\lambda _{k})_{k\in \mathbb{Z} }$ are independent copies of a random vector $(\xi ,\lambda )\in \mathbb{N} \times (0,1)$ and the mean $\mathbb{E} \xi $ is finite (moderate sparsity) we obtain stable limit laws for $X_{n}$, properly normalized and centered, as $n\to \infty $. While the case $\xi \leq M$ a.s. for some deterministic $M>0$ (weak sparsity) was analyzed by Matzavinos et al., the case $\mathbb{E} \xi =\infty $ (strong sparsity) will be analyzed in a forthcoming paper.

Lupu introduced a coupling between a random walk loop-soup and a Gaussian free field, where the sign of the field is constant on each cluster of loops. This coupling is a signed version of isomorphism theorems relating the square of the GFF to the occupation field of Markovian trajectories. His construction starts with a loop-soup, and by adding additional randomness samples a GFF out of it. In this article we provide the inverse construction: starting from a signed free field and using a self-interacting random walk related to this field, we construct a random walk loop-soup. Our construction relies on the previous work by Sabot and Tarrès, which inverts the coupling from the square of the GFF rather than the signed GFF itself.

Complex arithmetic random waves are stationary Gaussian complex-valued solutions of the Helmholtz equation on the two-dimensional flat torus. We use Wiener-Itô chaotic expansions in order to derive a complete characterization of the second order high-energy behaviour of the total number of phase singularities of these functions. Our main result is that, while such random quantities verify a universal law of large numbers, they also exhibit non-universal and non-central second order fluctuations that are dictated by the arithmetic nature of the underlying spectral measures. Such fluctuations are qualitatively consistent with the cancellation phenomena predicted by Berry (2002) in the case of complex random waves on compact planar domains. Our results extend to the complex setting recent pathbreaking findings by Rudnick and Wigman (2008), Krishnapur, Kurlberg and Wigman (2013) and Marinucci, Peccati, Rossi and Wigman (2016). The exact asymptotic characterization of the variance is based on a fine analysis of the Kac-Rice kernel around the origin, as well as on a novel use of combinatorial moment formulae for controlling long-range weak correlations. As a by-product of our analysis, we also deduce explicit bounds in smooth distances for the second order non-central results evoked above.

We establish two different, but related results for random walks in the domain of attraction of a stable law of index $\alpha $. The first result is a local large deviation upper bound, valid for $\alpha \in (0,1) \cup (1,2)$, which improves on the classical Gnedenko and Stone local limit theorems. The second result, valid for $\alpha \in (0,1)$, is the derivation of necessary and sufficient conditions for the random walk to satisfy the strong renewal theorem (SRT). This solves a long-standing problem, which dates back to the 1962 paper of Garsia and Lamperti [GL62] for renewal processes (i.e. random walks with non-negative increments), and to the 1968 paper of Williamson [Wil68] for general random walks.

We introduce a natural extension of the exclusion process to hypergraphs and prove an upper bound for its mixing time. In particular we show the existence of a constant $C$ such that for any connected, regular hypergraph $G$ within some natural class, the $\varepsilon $-mixing time of the exclusion process on $G$ with any feasible number of particles can be upper-bounded by $CT_{\text{EX} (2,G)}\log (|V|/\varepsilon )$, where $|V|$ is the number of vertices in $G$ and $T_{\text{EX} (2,G)}$ is the 1/4-mixing time of the corresponding exclusion process with just two particles. Moreover we show this is optimal in the sense that there exist hypergraphs in the same class for which $T_{\mathrm{EX} (2,G)}$ and the mixing time of just one particle are not comparable. The proofs involve an adaptation of the chameleon process, a technical tool invented by Morris ([14]) and developed by Oliveira ([15]) for studying the exclusion process on a graph.

We consider nearest neighbour spatial random permutations on $\mathbb Z ^{d}$. In this case, the energy of the system is proportional to the sum of all cycle lengths, and the system can be interpreted as an ensemble of edge-weighted, mutually self-avoiding loops. The constant of proportionality, $\alpha $, is the order parameter of the model. Our first result is that in a parameter regime of edge weights where it is known that a single self-avoiding loop is weakly space filling, long cycles of spatial random permutations are still exponentially unlikely. For our second result, we embed a self-avoiding walk into a background of spatial random permutations, and condition it to cover a macroscopic distance. For large values of $\alpha $ (where long cycles are very unlikely) we show that this walk collapses to a straight line in the scaling limit, and give bounds on the fluctuations that are almost sufficient for diffusive scaling. For proving our results, we develop the concepts of spatial strong Markov property and iterative sampling for spatial random permutations, which may be of independent interest. Among other things, we use them to show exponential decay of correlations for large values of $\alpha $ in great generality.

We show in this article that there exists no $H$-fractional Brownian field indexed by the cylinder $\mathbb{S} ^{1} \times ]0,\varepsilon [$ endowed with its product distance $d$ for any $\varepsilon >0$ and $H>0$. This is equivalent to say that $d^{2H}$ is not a negative definite kernel, which also leaves us without a proof that many classical stationary kernels, such that the Gaussian and exponential kernels, are positive definite kernels – or valid covariances – on the cylinder.

We generalise this result from the cylinder to any Riemannian Cartesian product with a minimal closed geodesic. We also investigate the case of the cylinder endowed with a distance asymptotically close to the product distance in the neighbourhood of a circle.

As a consequence of our result, we show that the set of $H$ such that $d^{2H}$ is negative definite behaves in a discontinuous way with respect to the Gromov-Hausdorff convergence on compact metric spaces.

These results extend our comprehension of kernel construction on metric spaces, and in particular call for alternatives to classical kernels to allow for Gaussian modelling and kernel method learning on cylinders.

In this paper, we show that the first passage time in the frog model on $\mathbb{Z} ^{d}$ with $d\geq 2$ has a sublinear variance. This implies that the central limit theorem does not hold at least with the standard diffusive scaling. The proof is based on the method introduced in [4, 11] combined with a control of the maximal weight of paths in a locally dependent site-percolation. We also apply this method to get the linearity of the lengths of optimal paths.

We consider a spin glass at temperature $T = 0$ where the underlying graph is a locally finite tree. We prove for a wide range of coupling distributions that uniqueness of ground states is equivalent to the maximal flow from any vertex to $\infty $ (where each edge $e$ has capacity $|J_{e}|$) being equal to zero which is equivalent to recurrence of the simple random walk on the tree.

Fragmentation processes are part of a broad class of models describing the evolution of a system of particles which split apart at random. These models are widely used in biology, materials science and nuclear physics, and their asymptotic behaviour at large times is interesting both mathematically and practically. The spine decomposition is a key tool in its study. In this work, we consider the class of compensated fragmentations, or homogeneous growth-fragmentations, recently defined by Bertoin. We give a complete spine decomposition of these processes in terms of a Lévy process with immigration, and apply our result to study the asymptotic properties of the derivative martingale.

In this paper we present new limit theorems for variational functionals of stationary increments Lévy driven moving averages in the high frequency setting. More specifically, we will show the “law of large numbers” and a “central limit theorem”, which heavily rely on the kernel, the driving Lévy process and the properties of the functional under consideration. The first order limit theory consists of three different cases. For one of the appearing limits, which we refer to as the ergodic type limit, we prove the associated weak limit theory, which again consists of three different cases. Our work is related to [10, 7], who considered power variation functionals of stationary increments Lévy driven moving averages. However, the asymptotic theory of the present paper is more complex. In particular, the weak limit theorems are derived for an arbitrary Appell rank of the involved functional.

We study the evolution of a random walker on a conservative dynamic random environment composed of independent particles performing simple symmetric random walks, generalizing results of [16] to higher dimensions and more general transition kernels without the assumption of uniform ellipticity or nearest-neighbour jumps. Specifically, we obtain a strong law of large numbers, a functional central limit theorem and large deviation estimates for the position of the random walker under the annealed law in a high density regime. The main obstacle is the intrinsic lack of monotonicity in higher-dimensional, non-nearest neighbour settings. Here we develop more general renormalization and renewal schemes that allow us to overcome this issue. As a second application of our methods, we provide an alternative proof of the ballistic behaviour of the front of (the discrete-time version of) the infection model introduced in [23].

We consider a general class of stochastic optimal control problems, where the state process lives in a real separable Hilbert space and is driven by a cylindrical Brownian motion and a Poisson random measure; no special structure is imposed on the coefficients, which are also allowed to be path-dependent; in addition, the diffusion coefficient can be degenerate. For such a class of stochastic control problems, we prove, by means of purely probabilistic techniques based on the so-called randomization method, that the value of the control problem admits a probabilistic representation formula (known as non-linear Feynman-Kac formula) in terms of a suitable backward stochastic differential equation. This probabilistic representation considerably extends current results in the literature on the infinite-dimensional case, and it is also relevant in finite dimension. Such a representation allows to show, in the non-path-dependent (or Markovian) case, that the value function satisfies the so-called randomized dynamic programming principle. As a consequence, we are able to prove that the value function is a viscosity solution of the corresponding Hamilton-Jacobi-Bellman equation, which turns out to be a second-order fully non-linear integro-differential equation in Hilbert space.

Let $a\in \mathbb{R} $ denote an unknown stationary target with a known distribution $\mu \in \mathcal{P(\mathbb {R}} )$, the space of probability measures on $\mathbb{R} $. A diffusive searcher $X(\cdot )$ sets out from the origin to locate the target. The time to locate the target is $T_{a}=\inf \{t\ge 0: X(t)=a\}$. The searcher has a given constant diffusion rate $D>0$, but its drift $b$ can be set by the search designer from a natural admissible class $\mathcal{D} _{\mu }$ of drifts. Thus, the diffusive searcher is a Markov process generated by the operator $L=\frac{D} {2}\frac{d^{2}} {dx^{2}}+b(x)\frac{d} {dx}$. For a given drift $b$, the expected time of the search is \[ \int _{\mathbb{R} } (E^{(b)}_{0}T_{a})\thinspace \mu (da).\tag{0.1} \] Our aim is to minimize this expected search time over all admissible drifts $b\in \mathcal{D} _{\mu }$. For measures $\mu $ that satisfy a certain balance condition between their restriction to the positive axis and their restriction to the negative axis, a condition satisfied, in particular, by all symmetric measures, we can give a complete answer to the problem. We calculate the above infimum explicitly, we classify the measures for which the infimum is attained, and in the case that it is attained, we calculate the minimizing drift explicitly. For measures that do not satisfy the balance condition, we obtain partial results.

We study a real-valued Lévy-type process $X$, which is locally $\alpha $-stable in the sense that its jump kernel is a combination of a ‘principal’ (state dependent) $\alpha $-stable part with a ‘residual’ lower order part. We show that under mild conditions on the local characteristics of a process (the jump kernel and the velocity field) the process is uniquely defined, is Markov, and has the strong Feller property. We approximate $X$ in law by a non-linear regression $\widetilde{X} ^{x}_{t}=\mathfrak{f} _{t}(x)+t^{1/\alpha }U^{x}_{t} $ with a deterministic regressor term $\mathfrak{f} _{t}(x)$ and $\alpha $-stable innovation term $U^{x}_{t}$, and provide error estimates for such an approximation. A case study is performed, revealing different types of assumptions which lead to various choices of regressor/innovation terms and various types of the estimates. The assumptions are quite general, cover the super-critical case $\alpha <1$, and allow non-symmetry of the Lévy kernel and unboundedness of the drift coefficient.

The stability properties of matrix-valued Riccati diffusions are investigated. The matrix-valued Riccati diffusion processes considered in this work are of interest in their own right, as a rather prototypical model of a matrix-valued quadratic stochastic process. Under rather natural observability and controllability conditions, we derive time-uniform moment and fluctuation estimates and exponential contraction inequalities. Our approach combines spectral theory with nonlinear semigroup methods and stochastic matrix calculus. This analysis seem to be the first of its kind for this class of matrix-valued stochastic differential equation. This class of stochastic models arise in signal processing and data assimilation, and more particularly in ensemble Kalman-Bucy filtering theory. In this context, the Riccati diffusion represents the flow of the sample covariance matrices associated with McKean-Vlasov-type interacting Kalman-Bucy filters. The analysis developed here applies to filtering problems with unstable signals.

We extend recent higher order concentration results in the discrete setting to include functions of possibly dependent variables whose distribution (on the product space) satisfies a logarithmic Sobolev inequality with respect to a difference operator that arises from Glauber type dynamics. Examples include the Ising model on a graph with $n$ sites with general, but weak interactions (i.e. in the Dobrushin uniqueness regime), for which we prove concentration results of homogeneous polynomials, as well as random permutations, and slices of the hypercube with dynamics given by either the Bernoulli-Laplace or the symmetric simple exclusion processes.

We study the random planar maps obtained from supercritical Galton–Watson trees by adding the horizontal connections between successive vertices at each level. These are the hyperbolic analog of the maps studied by Curien, Hutchcroft and Nachmias in [15], and a natural model of random hyperbolic geometry. We first establish metric hyperbolicity properties of these maps: we show that they admit bi-infinite geodesics and satisfy a weak version of Gromov-hyperbolicity. We also study the simple random walk on these maps: we identify their Poisson boundary and, in the case where the underlying tree has no leaf, we prove that the random walk has positive speed. Some of the methods used here are robust, and allow us to obtain more general results about planar maps containing a supercritical Galton–Watson tree.

We study variable-speed random walks on $\mathbb{Z} $ driven by a family of nearest-neighbor time-dependent random conductances $\{a_{t}(x,x+1)\colon x\in \mathbb{Z} ,\,t\ge 0\}$ whose law is assumed invariant and ergodic under space-time shifts. We prove a quenched invariance principle for the random walk under the minimal moment conditions on the environment; namely, assuming only that the conductances possess the first positive and negative moments. A novel ingredient is the representation of the parabolic coordinates and the corrector via a dual random walk which is considerably easier to analyze.

We establish several existence, uniqueness and comparison results for $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs with one and two continuous barriers under the assumptions that the generator $g$ satisfies a one-sided Osgood condition together with a very general growth condition in $y$, a uniform continuity condition and/or a sub-linear growth condition in $z$, and a generalized Mokobodzki condition for reflected BSDEs which relates the growth of $g$ and that of the barriers. This generalized Mokobodzki condition is proved to be necessary for existence of $L^{1}$ solutions of the reflected BSDEs. We also prove that the $L^{1}$ solutions of reflected BSDEs can be approximated by a penalization method and by some sequences of $L^{1}$ solutions of reflected BSDEs. The approach is based on a combination between existing methods, their refinement and perfection, but also on some novel ideas and techniques. These results strengthen some existing work on the $L^{1}$ solutions of non-reflected BSDEs and reflected BSDEs.

A sequence of large invertible matrices given by a small random perturbation around a fixed diagonal and positive matrix induces a random dynamics on a high-dimensional sphere. For a certain class of rotationally invariant random perturbations it is shown that the dynamics approaches the stable fixed points of the unperturbed matrix up to errors even if the strength of the perturbation is large compared to the relative increase of nearby diagonal entries of the unperturbed matrix specifying the local hyperbolicity.

We study the $\nabla \phi $ model with uniformly convex Hamiltonian $\mathcal{H} (\phi ) := \sum V(\nabla \phi )$ and prove a quantitative rate of convergence for the finite-volume surface tension as well as a quantitative rate estimate for the $L^{2}$-norm for the field subject to affine boundary condition. One of our motivations is to develop a new toolbox for studying this problem that does not rely on the Helffer-Sjöstrand representation. Instead, we make use of the variational formulation of the partition function, the notion of displacement convexity from the theory of optimal transport, and the recently developed theory of quantitative stochastic homogenization.

Disagreement percolation connects a Gibbs lattice gas and i.i.d. site percolation on the same lattice such that non-percolation implies uniqueness of the Gibbs measure. This work generalises disagreement percolation to the hard-sphere model and the Boolean model. Non-percolation of the Boolean model implies the uniqueness of the Gibbs measure and exponential decay of pair correlations and finite volume errors. Hence, lower bounds on the critical intensity for percolation of the Boolean model imply lower bounds on the critical activity for a (potential) phase transition. These lower bounds improve upon known bounds obtained by cluster expansion techniques. The proof uses a novel dependent thinning from a Poisson point process to the hard-sphere model, with the thinning probability related to a derivative of the free energy.

For a random walk killed at leaving a cone we suggest two new constructions of a positive harmonic function. These constructions allow one to remove a quite strong extendability assumption, which has been imposed in our previous paper (Denisov and Wachtel, 2015, Random walks in cones). As a consequence, all the limit results from that paper remain true for cones which are either convex or star-like and $C^{2}$.

We provide a direct proof of Cramér’s theorem for geodesic random walks in a complete Riemannian manifold $(M,g)$. We show how to exploit the vector space structure of the tangent spaces to study large deviation properties of geodesic random walks in $M$. Furthermore, we reveal the geometric obstructions one runs into.

To overcome these obstructions, we provide a Taylor expansion of the inverse Riemannian exponential map, together with appropriate bounds. Furthermore, we compare the differential of the Riemannian exponential map to parallel transport. Finally, we show how far geodesics, possibly starting in different points, may spread in a given amount of time.

With all geometric results in place, we obtain the analogue of Cramér’s theorem for geodesic random walks by showing that the curvature terms arising in this geometric analysis can be controlled and are negligible on an exponential scale.

Take a continuous-time Galton-Watson tree and pick $k$ distinct particles uniformly from those alive at a time $T$. What does their genealogical tree look like? The case $k=2$ has been studied by several authors, and the near-critical asymptotics for general $k$ appear in Harris, Johnston and Roberts (2018) [9]. Here we give the full picture.

In this paper, we establish a probabilistic representation for two integration by parts formulas, one being of Bismut-Elworthy-Li’s type, for the marginal law of a one-dimensional diffusion process killed at a given level. These formulas are established by combining a Markovian perturbation argument with a tailor-made Malliavin calculus for the underlying Markov chain structure involved in the probabilistic representation of the original marginal law. Among other applications, an unbiased Monte Carlo path simulation method for both integration by parts formula stems from the previous probabilistic representations.

We prove that the logarithm of the determinant of a Wigner matrix satisfies a central limit theorem in the limit of large dimension. Previous results about fluctuations of such determinants required that the first four moments of the matrix entries match those of a Gaussian [53]. Our work treats symmetric and Hermitian matrices with centered entries having the same variance and subgaussian tail. In particular, it applies to symmetric Bernoulli matrices and answers an open problem raised in [54]. The method relies on (1) the observable introduced in [9] and the stochastic advection equation it satisfies, (2) strong estimates on the Green function as in [12], (3) fixed energy universality [10], (4) a moment matching argument [52] using Green’s function comparison [21].

The goal of this paper is to define stochastic integrals and to solve stochastic differential equations for typical paths taking values in a possibly infinite dimensional separable Hilbert space without imposing any probabilistic structure. In the spirit of [33, 37] and motivated by the pricing duality result obtained in [4] we introduce an outer measure as a variant of the pathwise minimal superhedging price where agents are allowed to trade not only in $\omega $ but also in $\int \omega \,d\omega :=\omega ^{2} -\langle \omega \rangle $ and where they are allowed to include beliefs in future paths of the price process expressed by a prediction set. We then call a property to hold true on typical paths if the set of paths where the property fails is null with respect to our outer measure. It turns out that adding the second term $\omega ^{2} -\langle \omega \rangle $ in the definition of the outer measure enables to directly construct stochastic integrals which are continuous, even for typical paths taking values in an infinite dimensional separable Hilbert space. Moreover, when restricting to continuous paths whose quadratic variation is absolutely continuous with uniformly bounded derivative, a second construction of model-free stochastic integrals for typical paths is presented, which then allows to solve in a model-free way stochastic differential equations for typical paths.

This paper extends the link between stochastic approximation ($SA$) theory and randomized urn models developed in [32], and their applications to clinical trials introduced in [2, 3, 4]. We no longer assume that the drawing rule is uniform among the balls of the urn (which contains $d$ colors), but can be reinforced by a function $f$. This is a way to model risk aversion. Firstly, by considering that $f$ is concave or convex and by reformulating the dynamics of the urn composition as an $SA$ algorithm with remainder, we derive the $a.s.$ convergence and the asymptotic normality (Central Limit Theorem, $CLT$) of the normalized procedure by calling upon the so-called $ODE$ and $SDE$ methods. An in depth analysis of the case $d=2$ exhibits two different behaviors: a single equilibrium point when $f$ is concave, and, when $f$ is convex, a transition phase from a single attracting equilibrium to a system with two attracting and one repulsive equilibrium points. The last setting is solved using results on non-convergence toward noisy and noiseless “traps” in order to deduce the $a.s.$ convergence toward one of the attracting points. Secondly, the special case of a Pólya urn (when the addition rule is the $I_{d}$ matrix) is analyzed, still using result from $SA$ theory about “traps”. Finally, these results are used to solve another urn model with a more natural nonlinear drawing rule and we conclude by an example of application to optimal asset allocation in Finance.

We introduce a set of tools which simplify and streamline the proofs of limit theorems concerning near-critical particles in branching random walks under optimal assumptions. We exemplify our method by giving another proof of the Seneta-Heyde norming for the critical additive martingale, initially due to Aïdékon and Shi. The method involves in particular the replacement of certain second moment estimates by truncated first moment bounds, and the replacement of ballot-type theorems for random walks by estimates coming from an explicit expression for the potential kernel of random walks killed below the origin. Of independent interest might be a short, self-contained proof of this expression, as well as a criterion for convergence in probability of non-negative random variables in terms of conditional Laplace transforms.

We consider $d$ independent walkers in the same random environment in $ \mathbb{Z} $. Our assumption on the law of the environment is such that a single walker is transient to the right but subballistic. We show that — no matter what $d$ is — the $d$ walkers meet infinitely often, i.e. there are almost surely infinitely many times for which all the random walkers are at the same location.

We consider the so-called Dickman subordinator, whose Lévy measure has density $\frac{1} {x}$ restricted to the interval $(0,1)$. The marginal density of this process, known as the Dickman function, appears in many areas of mathematics, from number theory to combinatorics. In this paper, we study renewal processes in the domain of attraction of the Dickman subordinator, for which we prove local renewal theorems. We then present applications to marginally relevant disordered systems, such as pinning and directed polymer models, and prove sharp second moment estimates on their partition functions.

We show that the pointed measured Gromov convergence of the underlying spaces implies (or under some condition, is equivalent to) the weak convergence of Brownian motions under Riemannian Curvature-Dimension conditions.

Consider a continuous-state branching population constructed as a flow of nested subordinators. Inverting the subordinators and reversing time give rise to a flow of coalescing Markov processes with negative jumps, which correspond to the ancestral lineages of individuals in the current generation. The process of the ancestral lineage of a fixed individual is the Siegmund dual process of the continuous-state branching process. We study its semi-group, its long-term behaviour and its generator. In order to follow the coalescences in the ancestral lineages and to describe the backward genealogy of the population, we define non-exchangeable Markovian coalescent processes obtained by sampling individuals according to an independent Poisson point process over the flow. These coalescent processes are called consecutive coalescents, as only consecutive blocks can merge. They are characterized in law by finite measures on $\mathbb{N} $ which can be thought as the offspring distributions of some inhomogeneous immortal Galton-Watson processes forward in time.

We consider an SOS (solid-on-solid) model, with spin values from the set of all integers, on a Cayley tree of order $k\geq 2$ and are interested in tree-automorphism invariant gradient Gibbs measures (GGMs) of the model. Such a measure corresponds to a boundary law (a function defined on vertices of the Cayley tree) satisfying a functional equation. In the ferromagnetic SOS case on the binary tree we find up to five solutions to a class of period-$4$ height-periodic boundary law equations (in particular, some period-2 height-periodic ones). We show that these boundary laws define up to four distinct GGMs. Moreover, we construct some period-$3$ height-periodic boundary laws on the Cayley tree of arbitrary order $k\geq 2$, which define GGMs different from the $4$-periodic ones.

We consider a $d$-dimensional branching particle system in a random environment. Suppose that the initial measures converge weakly to a measure with bounded density. Under the Mytnik-Sturm branching mechanism, we prove that the corresponding empirical measure $X_{t}^{n}$ converges weakly in the Skorohod space $D([0,T];M_{F}(\mathbb{R} ^{d}))$ and the limit has a density $u_{t}(x)$, where $M_{F}(\mathbb{R} ^{d})$ is the space of finite measures on $\mathbb{R} ^{d}$. We also derive a stochastic partial differential equation $u_{t}(x)$ satisfies. By using the techniques of Malliavin calculus, we prove that $u_{t}(x)$ is jointly Hölder continuous in time with exponent $\frac{1} {2}-\epsilon $ and in space with exponent $1-\epsilon $ for any $\epsilon >0$.

In this article, we consider the stochastic wave equation on $\mathbb{R} _{+} \times \mathbb{R} $, driven by a linear multiplicative space-time homogeneous Gaussian noise whose temporal and spatial covariance structures are given by locally integrable functions $\gamma $ (in time) and $f$ (in space), which are the Fourier transforms of tempered measures $\nu $ on $\mathbb{R} $, respectively $\mu $ on $\mathbb{R} $. Our main result shows that the law of the solution $u(t,x)$ of this equation is absolutely continuous with respect to the Lebesgue measure.

We prove estimates for $\mathbb{E} \| X: \ell _{p'}^{n} \to \ell _{q}^{m}\|$ for $p,q\ge 2$ and any random matrix $X$ having the entries of the form $a_{ij}Y_{ij}$, where $Y=(Y_{ij})_{1\le i\le m, 1\le j\le n}$ has i.i.d. isotropic log-concave rows and $p'$ denotes the Hölder conjugate of $p$. This generalises a result of Guédon, Hinrichs, Litvak, and Prochno for Gaussian matrices with independent entries. Our estimate is optimal up to logarithmic factors. As a byproduct we provide an analogous bound for $m\times n$ random matrices, whose entries form an unconditional vector in $\mathbb{R} ^{mn}$. We also prove bounds for norms of matrices whose entries are certain Gaussian mixtures.

We consider an extension of the interchange process on the complete graph, in which a fraction of the transpositions are replaced by ‘reversals’. The model is motivated by statistical physics, where it plays a role in stochastic representations of xxz-models. We prove convergence to PD($\tfrac{1} {2}$) of the rescaled cycle sizes, above the critical point for the appearance of macroscopic cycles. This extends a result of Schramm on convergence to PD(1) for the usual interchange process.

We establish decomposition formulas for nonnegative infinitely divisible processes. They allow to give an explicit expression of their Lévy measure. In the special case of infinitely divisible permanental processes, one of these decompositions represents a new isomorphism theorem involving the local time process of a transient Markov process. We obtain in this case the expression of the Lévy measure of the total local time process which is in itself a new result on the local time process. Finally, we identify a determining property of the local times for their connection with permanental processes.

It has been recently understood [9, 24, 30] that for a general class of percolation models on $\mathbb{Z} ^{d}$ satisfying suitable decoupling inequalities, which includes i.a. Bernoulli percolation, random interlacements and level sets of the Gaussian free field, large scale geometry of the unique infinite cluster in strongly percolative regime is qualitatively the same; in particular, the random walk on the infinite cluster satisfies the quenched invariance principle, Gaussian heat-kernel bounds and local CLT.

In this paper we consider the random walk loop soup on $\mathbb{Z} ^{d}$ in dimensions $d\geq 3$. An interesting aspect of this model is that despite its similarity and connections to random interlacements and the Gaussian free field, it does not fall into the above mentioned general class of percolation models, since the required decoupling inequalities are not valid.

We identify weaker (and more natural) decoupling inequalities and prove that (a) they do hold for the random walk loop soup and (b) all the results about the large scale geometry of the infinite percolation cluster proved for the above mentioned class of models hold also for models that satisfy the weaker decoupling inequalities. Particularly, all these results are new for the vacant set of the random walk loop soup. (The range of the random walk loop soup has been addressed by Chang [6] by a model specific approximation method, which does not apply to the vacant set.)

Finally, we prove that the strongly supercritical regime for the vacant set of the random walk loop soup is non-trivial. It is expected, but open at the moment, that the strongly supercritical regime coincides with the whole supercritical regime.

In this work, we consider the loop-erased random walk (LERW) in three dimensions and give asymptotic estimates for the one-point function of LERW and the non-intersection probability of LERW and simple random walk for dyadic scales. These estimates will be crucial to the characterization of the convergence of LERW to its scaling limit in natural parametrization. As a step in the proof, we also obtain a coupling of two pairs of LERW and SRW with different starting points conditioned to avoid each other.

The frog model is an infection process in which dormant particles begin moving and infecting others once they become infected. We show that on the rooted $d$-ary tree with particle density $\Omega (d^{2})$, the set of visited sites contains a linearly expanding ball and the number of visits to the root grows linearly with high probability.

Lemma 4.8 in the article [1] contains a mistake, which implies a weaker regularity estimate than the one stated in Proposition 4.11. This does not affect the proof of Theorem 2.1, but Theorems 2.2 and 2.3 only follow from the given proof if either the space dimension $d$ is equal to $2$, or the nonlinearity $F(U,V)$ is linear in $V$. To fix this problem and provide a proof of Theorems 2.2 and 2.3 valid in full generality, we consider an alternative formulation of the fixed-point problem, involving a modified integration operator with nonlocal singularity and a slightly different regularity structure. We provide the multilevel Schauder estimates and renormalisation-group analysis required for the fixed-point argument in this new setting.

We establish phase transitions for continuum Delaunay multi-type particle systems (continuum Potts or Widom-Rowlinson models) with a repulsive interaction between particles of different types. Our interaction potential depends solely on the length of the Delaunay edges. We show that a phase transition occurs for sufficiently large activities and for sufficiently large potential parameter proving an old conjecture of Lebowitz and Lieb extended to the Delaunay structure. Our approach involves a Delaunay random-cluster representation analogous to the Fortuin-Kasteleyn representation of the Potts model. The phase transition manifests itself in the mixed site-bond percolation of the corresponding random-cluster model. Our proofs rely mainly on geometric properties of Delaunay tessellations in $\mathbb{R} ^{2} $ and on recent studies [DDG12] of Gibbs measures for geometry-dependent interactions. The main tool is a uniform bound on the number of connected components in the Delaunay graph which provides a novel approach to Delaunay Widom Rowlinson models based on purely geometric arguments. The interaction potential ensures that shorter Delaunay edges are more likely to be open and thus offsets the possibility of having an unbounded number of connected components.

We compute the characteristic functional of the Dirichlet–Ferguson measure over a locally compact Polish space and prove continuous dependence of the random measure on the parameter measure. In finite dimension, we identify the dynamical symmetry algebra of the characteristic functional of the Dirichlet distribution with a simple Lie algebra of type $A$. We study the lattice determined by characteristic functionals of categorical Dirichlet posteriors, showing that it has a natural structure of weight Lie algebra module and providing a probabilistic interpretation. A partial generalization to the case of the Dirichlet–Ferguson measure is also obtained.

In recent years, a number of functional inequalities have been derived for Poisson random measures, with a wide range of applications. In this paper, we prove that such inequalities can be extended to the setting of marked temporal point processes, under mild assumptions on their Papangelou conditional intensity. First, we derive a Poincaré inequality. Second, we prove two transportation cost inequalities. The first one refers to functionals of marked point processes with a Papangelou conditional intensity and is new even in the setting of Poisson random measures. The second one refers to the law of marked temporal point processes with a Papangelou conditional intensity, and extends a related inequality which is known to hold on a general Poisson space. Finally, we provide a variational representation of the Laplace transform of functionals of marked point processes with a Papangelou conditional intensity. The proofs make use of an extension of the Clark-Ocone formula to marked temporal point processes. Our results are shown to apply to classes of renewal, nonlinear Hawkes and Cox point processes.

We prove existence and uniqueness of distributional solutions to the KPZ equation globally in space and time, with techniques from paracontrolled analysis. Our main tool for extending the analysis on the torus to the full space is a comparison result that gives quantitative upper and lower bounds for the solution. We then extend our analysis to provide a path-by-path construction of the random directed polymer measure on the real line and we derive a variational characterisation of the solution to the KPZ equation.

The main result of this paper is that there are examples of stochastic partial differential equations [hereforth, SPDEs] of the type \[ \partial _{t} u=\tfrac{1} {2}\Delta u +\sigma (u)\eta \qquad \text{on $(0\,,\infty )\times \mathbb {R}^{3}$} \] such that the solution exists and is unique as a random field in the sense of Dalang [6] and Walsh [31], yet the solution has unbounded oscillations in every open neighborhood of every space-time point. We are not aware of the existence of such a construction in spatial dimensions below $3$.

En route, it will be proved that when $\sigma (u)=u$ there exist a large family of parabolic SPDEs whose moment Lyapunov exponents grow at least sub exponentially in its order parameter in the sense that there exist $A_{1},\beta \in (0\,,1)$ such that \[ \underline{\gamma } (k) := \liminf _{t\to \infty }t^{-1}\inf _{x\in \mathbb{R} ^{3}} \log \mathrm{E} \left (|u(t\,,x)|^{k}\right ) \geqslant A_{1}\exp (A_{1} k^{\beta }) \qquad \text{for all $k\geqslant 2$} . \] This sort of “super intermittency” is combined with a local linearization of the solution, and with techniques from Gaussian analysis in order to establish the unbounded oscillations of the sample functions of the solution to our SPDE.

Asymptotic expansion of the distribution of a perturbation $Z_n$ of a Skorohod integral jointly with a reference variable $X_n$ is derived. We introduce a second-order interpolation formula in frequency domain to expand a characteristic functional and combine it with the scheme developed in the martingale expansion. The second-order interpolation and Fourier inversion give asymptotic expansion of the expectation $E[f(Z_n,X_n)]$ for differentiable functions $f$ and also measurable functions $f$. In the latter case, the interpolation method connects the two non-degeneracies of variables for finite $n$ and $\infty $. Random symbols are used for expressing the asymptotic expansion formula. Quasi tangent, quasi torsion and modified quasi torsion are introduced in this paper. We identify these random symbols for a certain quadratic form of a fractional Brownian motion and for a quadratic from of a fractional Brownian motion with random weights. For a quadratic form of a Brownian motion with random weights, we observe that our formula reproduces the formula originally obtained by the martingale expansion.

This paper is centered on the random graph generated by a Doeblin-type coupling of discrete time processes on a countable state space whereby when two paths meet, they merge. This random graph is studied through a novel subgraph, called a bridge graph, generated by paths started in a fixed state at any time. The bridge graph is made into a unimodular network by marking it and selecting a root in a specified fashion. The unimodularity of this network is leveraged to discern global properties of the larger Doeblin graph. Bi-recurrence, i.e., recurrence both forwards and backwards in time, is introduced and shown to be a key property in uniquely distinguishing paths in the Doeblin graph, and also a decisive property for Markov chains indexed by $\mathbb{Z} $. Properties related to simulating the bridge graph are also studied.

The branching-ruin number of a tree, which describes its asymptotic growth and geometry, can be seen as a polynomial version of the branching number. This quantity was defined by Collevecchio, Kious and Sidoravicius (2018) in order to understand the phase transitions of the once-reinforced random walk (ORRW) on trees. Strikingly, this number was proved to be equal to the critical parameter of ORRW on trees.

In this paper, we continue the investigation of the link between the branching-ruin number and the criticality of random processes on trees.

First, we study random walks on random conductances on trees, when the conductances have an heavy tail at $0$, parametrized by some $p>1$, where $1/p$ is the exponent of the tail. We prove a phase transition recurrence/transience with respect to $p$ and identify the critical parameter to be equal to the branching-ruin number of the tree.

Second, we study a multi-excited random walk on trees where each vertex has $M$ cookies and each cookie has an infinite strength towards the root. Here again, we prove a phase transition recurrence/transience and identify the critical number of cookies to be equal to the branching-ruin number of the tree, minus 1. This result extends a conjecture of Volkov (2003). Besides, we study a generalized version of this process and generalize results of Basdevant and Singh (2009).

We investigate a class of models for opinion dynamics in a population with two interacting families of individuals. Each family has an intrinsic mean field “Voter-like” dynamics which is influenced by interaction with the other family. The interaction terms describe a cooperative/conformist or competitive/nonconformist attitude of one family with respect to the other. We prove chaos propagation, i.e., we show that on any time interval $[0,T]$, as the size of the system goes to infinity, each individual behaves independently of the others with transition rates driven by a macroscopic equation. We focus in particular on models with Lotka-Volterra type interactions, i.e., models with cooperative vs. competitive families. For these models, although the microscopic system is driven a.s. to consensus within each family, a periodic behaviour arises in the macroscopic scale.

In order to describe fluctuations between the limiting periodic orbits, we identify a slow variable in the microscopic system and, through an averaging principle, we find a diffusion which describes the macroscopic dynamics of such variable on a larger time scale.

In this paper, we prove the edge universality of largest eigenvalues for separable covariance matrices of the form $\mathcal{Q} :=A^{1/2}XBX^{*}A^{1/2}$. Here $X=(x_{ij})$ is an $n\times N$ random matrix with $x_{ij}=N^{-1/2}q_{ij}$, where $q_{ij}$ are $i.i.d.$ random variables with zero mean and unit variance, and $A$ and $B$ are respectively $n \times n$ and $N\times N$ deterministic non-negative definite symmetric (or Hermitian) matrices. We consider the high-dimensional case, i.e. ${n}/{N}\to d \in (0, \infty )$ as $N\to \infty $. Assuming $\mathbb{E} q_{ij}^{3}=0$ and some mild conditions on $A$ and $B$, we prove that the limiting distribution of the largest eigenvalue of $\mathcal{Q} $ coincide with that of the corresponding Gaussian ensemble (i.e. $\mathcal{Q} $ with $X$ being an $i.i.d.$ Gaussian matrix) as long as we have $\lim _{s \rightarrow \infty }s^{4} \mathbb{P} (\vert q_{ij} \vert \geq s)=0$, which is a sharp moment condition for edge universality. If we take $B=I$, then $\mathcal{Q} $ becomes the normal sample covariance matrix and the edge universality holds true without the vanishing third moment condition. So far, this is the strongest edge universality result for sample covariance matrices with correlated data (i.e. non-diagonal $A$) and heavy tails, which improves the previous results in [6, 39] (assuming high moments and diagonal $A$), [37] (assuming high moments) and [14] (assuming diagonal $A$).

In this paper, we establish concentration inequalities both for functionals of the whole solution on an interval $[0,T]$ of an additive SDE driven by a fractional Brownian motion with Hurst parameter $H\in (0,1)$ and for functionals of discrete-time observations of this process. Then, we apply this general result to specific functionals related to discrete and continuous-time occupation measures of the process.

The hiring problem is studied for general strategies based only on the relative ranking of the candidates; this includes some well known strategies studied before such as hiring above the median. We give general limit theorems for the number of hired candidates and some other properties, extending previous results. The results exhibit a dichotomy between two classes of rank-based strategies: either the asymptotics of the process are determined by the early events, with a.s. convergence of suitably normalized random variables, or there is a mixing behaviour without long-term memory and with asymptotic normality.

Consider tossing a collection of coins, each fair or biased towards heads, and take the distribution of the total number of heads that result. It is natural to suppose that this distribution should be ‘more random’ when each coin is fairer. In this paper, we prove a 40 year old conjecture of Shepp and Olkin, by showing that the Shannon entropy is monotonically increasing in this case, using a construction inspired by optimal transport theory. We discuss whether this result can be generalized to $q$-Rényi and $q$-Tsallis entropies, for a range of values of $q$.

We give a new criterion for ballistic behavior of random walks in random environments which are low disorder perturbations of the simple symmetric random walk on $\mathbb{Z} ^{d}$, for $d\geq 2$. This extends the results from 2003 established by Sznitman in [12] and, in particular, allow us to give new examples of ballistic RWREs in dimension $d=3$ which do not satisfy Kalikow’s condition, through a new sharp version of Kalikow’s criteria. Essentially, this new criterion states that ballisticity occurs whenever the average local drift of the walk is not too small when compared to the standard deviation of the environment. Its proof relies on applying coarse-graining methods together with a variation of the Azuma-Hoeffding concentration inequality in order to verify the fulfillment of a ballisticity condition by Berger, Drewitz and Ramírez.

This work explores and develops elements of Stein’s method of approximation in the infinitely divisible setting, and its connections to functional analysis. It is mainly concerned with multivariate self-decomposable laws without finite first moment and, in particular, with $\alpha $-stable ones, $\alpha \in (0,1]$. At first, several characterizations of these laws via covariance identities are presented. In turn, these characterizations lead to integro-differential equations which are solved with the help of both semigroup and Fourier methodologies. Then, Poincaré-type inequalities for self-decomposable laws having finite first moment are revisited. In this non-local setting, several algebraic quantities (such as the carré du champs and its iterates) originating in the theory of Markov diffusion operators are computed. Finally, rigidity and stability results for the Poincaré-ratio functional of the rotationally invariant $\alpha $-stable laws, $\alpha \in (1,2)$, are obtained; and as such they recover the classical Gaussian setting as $\alpha \to 2$.

In this paper, we established a quadratic transportation cost inequality under the uniform/maximum norm for solutions of stochastic heat equations driven by multiplicative space-time white noise. The proof is based on a new inequality we obtained for the moments of the stochastic convolution with respect to space-time white noise, which is of independent interest. The solutions of such stochastic partial differential equations are typically not semimartingales on the state space.

Given a vector $F =(F_{1},\ldots ,F_{m})$ of Poisson functionals $F_{1},\ldots ,F_{m}$, we investigate the proximity between $F$ and an $m$-dimensional centered Gaussian random vector $N_{\Sigma }$ with covariance matrix $\Sigma \in \mathbb{R} ^{m\times m}$. Apart from finding proximity bounds for the $d_{2}$- and $d_{3}$-distances, based on classes of smooth test functions, we obtain proximity bounds for the $d_{convex}$-distance, based on the less tractable test functions comprised of indicators of convex sets. The bounds for all three distances are shown to be of the same order, which is presumably optimal. The bounds are multivariate counterparts of the univariate second order Poincaré inequalities and, as such, are expressed in terms of integrated moments of first and second order difference operators. The derived second order Poincaré inequalities for indicators of convex sets are made possible by a new bound on the second derivatives of the solution to the Stein equation for the multivariate normal distribution. We present applications to the multivariate normal approximation of first order Poisson integrals and of statistics of Boolean models.

In this paper we prove a discretized version of Krylov’s estimate for discretized Itô processes. As applications, we study the weak and strong convergences for Euler’s approximation of mean-field SDEs with measurable discontinuous and linear growth coefficients. Moreover, we also show the propagation of chaos for Euler’s approximation of mean-field SDEs.

We consider a broad class of semilinear SPDEs with multiplicative noise driven by a finite-dimensional Wiener process. We show that, provided that an infinite-dimensional analogue of Hörmander’s bracket condition holds, the Malliavin matrix of the solution is an operator with dense range. In particular, we show that the laws of finite-dimensional projections of such solutions admit smooth densities with respect to Lebesgue measure. The main idea is to develop a robust pathwise solution theory for such SPDEs using rough paths theory, which then allows us to use a pathwise version of Norris’s lemma to work directly on the Malliavin matrix, instead of the “reduced Malliavin matrix” which is not available in this context. On our way of proving this result, we develop some new tools for the theory of rough paths like a rough Fubini theorem and a deterministic mild Itô formula for rough PDEs.

We present Rosenthal-type moment inequalities for matrix-valued U-statistics of order 2. As a corollary, we obtain new matrix concentration inequalities for U-statistics. One of our main technical tools, a version of the non-commutative Khintchine inequality for the spectral norm of the Rademacher chaos, could be of independent interest.

$\Lambda $-coalescents model genealogies of samples of individuals from a large population by means of a family tree. The tree’s leaves represent the individuals, and the lengths of the adjacent edges indicate the individuals’ time durations up to some common ancestor. These edges are called external branches. We consider typical external branches under the broad assumption that the coalescent has no dust component and maximal external branches under further regularity assumptions. As it transpires, the crucial characteristic is the coalescent’s rate of decrease $\mu (b)$, $b\geq 2$. The magnitude of a typical external branch is asymptotically given by $n/\mu (n)$, where $n$ denotes the sample size. This result, in addition to the asymptotic independence of several typical external lengths, holds in full generality, while convergence in distribution of the scaled external lengths requires that $\mu (n)$ is regularly varying at infinity. For the maximal lengths, we distinguish two cases. Firstly, we analyze a class of $\Lambda $-coalescents coming down from infinity and with regularly varying $\mu $. Here, the scaled external lengths behave as the maximal values of $n$ i.i.d. random variables, and their limit is captured by a Poisson point process on the positive real line. Secondly, we turn to the Bolthausen-Sznitman coalescent, where the picture changes. Now, the limiting behavior of the normalized external lengths is given by a Cox point process, which can be expressed by a randomly shifted Poisson point process.

Constrained Markov processes, such as reflecting diffusions, behave as an unconstrained process in the interior of a domain but upon reaching the boundary are controlled in some way so that they do not leave the closure of the domain. In this paper, the behavior in the interior is specified by a generator of a Markov process, and the constraints are specified by a controlled generator. Together, the generators define a constrained martingale problem. The desired constrained processes are constructed by first solving a simpler controlled martingale problem and then obtaining the desired process as a time-change of the controlled process. As for ordinary martingale problems, it is rarely obvious that the process constructed in this manner is unique. The primary goal of the paper is to show that from among the processes constructed in this way one can “select”, in the sense of Krylov, a strong Markov process. Corollaries to these constructions include the observation that uniqueness among strong Markov solutions implies uniqueness among all solutions. These results provide useful tools for proving uniqueness for constrained processes including reflecting diffusions. The constructions also yield viscosity semisolutions of the resolvent equation and, if uniqueness holds, a viscosity solution, without proving a comparison principle. We illustrate our results by applying them to reflecting diffusions in piecewise smooth domains. We prove existence of a strong Markov solution to the SDE with reflection, under conditions more general than in [13]: In fact our conditions are known to be optimal in the case of simple, convex polyhedrons with constant direction of reflection on each face ([10]). We also indicate how the results can be applied to processes with Wentzell boundary conditions and nonlocal boundary conditions.

In this paper linear stochastic transport and continuity equations with drift in critical $L^{p}$ spaces are considered. In this situation noise prevents shocks for the transport equation and singularities in the density for the continuity equation, starting from smooth initial conditions. Specifically, we first prove a result of Sobolev regularity of solutions, which is false for the corresponding deterministic equation. The technique needed to reach the critical case is new and based on parabolic equations satisfied by moments of first derivatives of the solution, opposite to previous works based on stochastic flows. The approach extends to higher order derivatives under more regularity of the drift term. By a duality approach, these regularity results are then applied to prove uniqueness of weak solutions to linear stochastic continuity and transport equations and certain well-posedness results for the associated stochastic differential equation (sDE) (roughly speaking, existence and uniqueness of flows and their $C^{\alpha }$ regularity, strong uniqueness for the sDE when the initial datum has diffuse law). Finally, we show two types of examples: on the one hand, we present well-posed sDEs, when the corresponding ODEs are ill-posed, and on the other hand, we give a counterexample in the supercritical case.

Consider a randomly-oriented two dimensional Manhattan lattice where each horizontal line and each vertical line is assigned, once and for all, a random direction by flipping independent and identically distributed coins. A deterministic walk is then started at the origin and at each step moves diagonally to the nearest vertex in the direction of the horizontal and vertical lines of the present location. This definition can be generalized, in a natural way, to larger dimensions, but we mainly focus on the two dimensional case. In this context the process localizes on two vertices at all large times, almost surely. We also provide estimates for the tail of the length of paths, when the walk is defined on the two dimensional lattice. In particular, the probability of the path to be larger than $n$ decays sub-exponentially in $n$. It is easy to show that higher dimensional paths may not localize on two vertices but will still eventually become periodic, and are therefore bounded.

A stationary stochastic geometric model is proposed for analyzing the data compression method used in one-bit compressed sensing. The data set is an unconstrained stationary set, for instance all of $\mathbb{R} ^{n}$ or a stationary Poisson point process in $\mathbb{R} ^{n}$. It is compressed using a stationary and isotropic Poisson hyperplane tessellation, assumed independent of the data. That is, each data point is compressed using one bit with respect to each hyperplane, which is the side of the hyperplane it lies on. This model allows one to determine how the intensity of the hyperplanes must scale with the dimension $n$ to ensure sufficient separation of different data by the hyperplanes as well as sufficient proximity of the data compressed together. The results have direct implications in compressed sensing and in source coding.

The infinite two-sided loop-erased random walk (LERW) is a measure on infinite self-avoiding walks that can be viewed as giving the law of the “middle part” of an infinite LERW loop going through $0$ and $\infty $. In this note we derive expressions for transition probabilities for this model in dimensions $d \ge 2$. For $d=2$ the formula can be further expressed in terms of a Laplacian with signed weights acting on certain discrete harmonic functions at the tips of the walk, and taking a determinant. The discrete harmonic functions are closely related to a discrete version of $z \mapsto \sqrt{z} $.

We consider the problem of optimally stopping a general one-dimensional stochastic differential equation (SDE) with generalised drift over an infinite time horizon. First, we derive a complete characterisation of the solution to this problem in terms of variational inequalities. In particular, we prove that the problem’s value function is the difference of two convex functions and satisfies an appropriate variational inequality in the sense of distributions. We also establish a verification theorem that is the strongest one possible because it involves only the optimal stopping problem’s data. Next, we derive the complete explicit solution to the problem that arises when the state process is a skew geometric Brownian motion and the reward function is the one of a financial call option. In this case, we show that the optimal stopping strategy can take several qualitatively different forms, depending on parameter values. Furthermore, the explicit solution to this special case shows that the so-called “principle of smooth fit” does not hold in general for optimal stopping problems involving solutions to SDEs with generalised drift.

In this paper, we study the asymptotic behavior of a supercritical $(\xi ,\psi )$-superprocess $(X_{t})_{t\geq 0}$ whose underlying spatial motion $\xi $ is an Ornstein-Uhlenbeck process on $\mathbb{R} ^{d}$ with generator $L = \frac{1} {2}\sigma ^{2}\Delta - b x \cdot \nabla $ where $\sigma , b >0$; and whose branching mechanism $\psi $ satisfies Grey’s condition and a perturbation condition which guarantees that, when $z\to 0$, $\psi (z)=-\alpha z + \eta z^{1+\beta } (1+o(1))$ with $\alpha > 0$, $\eta >0$ and $\beta \in (0, 1)$. Some law of large numbers and $(1+\beta )$-stable central limit theorems are established for $(X_{t}(f) )_{t\geq 0}$, where the function $f$ is assumed to be of polynomial growth. A phase transition arises for the central limit theorems in the sense that the forms of the central limit theorem are different in three different regimes corresponding to the branching rate being relatively small, large or critical at a balanced value.

We derive scaling limit results for the Hierarchical Random Hopping Dynamics for the cascading two-level GREM at low temperature at extreme time scales. It is known that in the cascading regime there are two static critical temperatures. We show that there exists a (narrow) set of fine tuning temperatures; when they lie below the static lowest critical temperature, three distinct dynamical phases emerge below the lowest critical temperature, with three different types of limiting dynamics depending on whether the temperature is (well) above or below, or at a fine tuning temperature, all of which are given in terms of K processes. We also derive scaling limit results for temperatures between the lowest and he highest critical ones, as well as aging results for all the limiting processes mentioned above, by taking a second small time limit.

We investigate the macroscopic energy diffusion of a disordered harmonic chain of oscillators, whose hamiltonian dynamics is perturbed by a degenerate conservative noise. After rescaling space and time diffusively, we prove that the equilibrium energy fluctuations evolve according to a linear heat equation. The diffusion coefficient is obtained from Varadhan’s non-gradient approach, and is equivalently defined through the Green-Kubo formula. Since the perturbation is very degenerate and the symmetric part of the generator does not have a spectral gap, the standard non-gradient method is reviewed under new perspectives.

Based on the rigorous path integral formulation of Liouville Conformal Field Theory initiated by David-Kupiainen-Rhodes-Vargas [6] on the Riemann sphere and David-Rhodes-Vargas [11] on the torus of modulus $\tau $, we give the exact asymptotic behaviour of the 1-point toric correlation function as $\mathrm{Im} \tau \to \infty $.

In agreement with formulae predicted within the bootstrap formalism of theoretical physics, our results feature an $(\mathrm{Im} \tau )^{-3/2}$ decay rate and we identify the derivative of DOZZ formula in the limit.

We study arithmetic progressions $\{a,a+b,a+2b,\dots ,a+(\ell -1) b\}$, with $\ell \ge 3$, in random subsets of the initial segment of natural numbers $[n]:=\{1,2,\dots , n\}$. Given $p\in [0,1]$ we denote by $[n]_{p}$ the random subset of $[n]$ which includes every number with probability $p$, independently of one another. The focus lies on sparse random subsets, i.e. when $p=p(n)=o(1)$ as $n\to +\infty $. Let $X_{\ell }$ denote the number of distinct arithmetic progressions of length $\ell $ which are contained in $[n]_{p}$. We determine the limiting distribution for $X_{\ell }$ not only for fixed $\ell \ge 3$ but also when $\ell =\ell (n)\to +\infty $ with $\ell =o(\log n)$. The main result concerns the joint distribution of the pair $(X_{\ell },X_{\ell '})$, $\ell >\ell '$, for which we prove a bivariate central limit theorem for a wide range of $p$. Interestingly, the question of whether the limiting distribution is trivial, degenerate, or non-trivial is characterised by the asymptotic behaviour (as $n\to +\infty $) of the threshold function $\psi _{\ell }=\psi _{\ell }(n):=np^{\ell -1}\ell $. The proofs are based on the method of moments and combinatorial arguments, such as an algorithmic enumeration of collections of arithmetic progressions.

A two-type version of the frog model on $\mathbb{Z} ^{d}$ is formulated, where active type $i$ particles move according to lazy random walks with probability $p_{i}$ of jumping in each time step ($i=1,2$). Each site is independently assigned a random number of particles. At time 0, the particles at the origin are activated and assigned type 1 and the particles at one other site are activated and assigned type 2, while all other particles are sleeping. When an active type $i$ particle moves to a new site, any sleeping particles there are activated and assigned type $i$, with an arbitrary tie-breaker deciding the type if the site is hit by particles of both types in the same time step. Let $G_{i}$ denote the event that type $i$ activates infinitely many particles. We show that the events $G_{1}\cap G_{2}^{\mathrm{c} }$ and $G_{1}^{\mathrm{c} }\cap G_{2}$ both have positive probability for all $p_{1},p_{2}\in (0,1]$. Furthermore, if $p_{1}=p_{2}$, then the types can coexist in the sense that the event $G_{1}\cap G_{2}$ has positive probability. We also formulate several open problems. For instance, we conjecture that, when the initial number of particles per site has a heavy tail, the types can coexist also when $p_{1}\neq p_{2}$.

We use jump processes with stochastic intensity to construct a class of reflection laws for billiard processes in the unit interval whose stationary distribution for the billiard position and its velocity is the product of the uniform distribution and the standard normal distribution. These billiard processes have Markovian reflection laws, meaning their velocity is constant between reflections but changes in a Markovian way at reflection times.