We present simple conditions under which the limiting genealogical process associated with a class of interacting particle systems with non-neutral selection mechanisms, as the number of particles grows, is a time-rescaled Kingman coalescent. Sequential Monte Carlo algorithms are popular methods for approximating integrals in problems such as non-linear filtering and smoothing which employ this type of particle system. Their performance depends strongly on the properties of the induced genealogical process. We verify the conditions of our main result for standard sequential Monte Carlo algorithms with a broad class of low-variance resampling schemes, as well as for conditional sequential Monte Carlo with multinomial resampling.

We study the asymptotic behavior as $t \to \infty $ of a time-dependent family $(\mu _{t})_{t \geq 0}$ of probability measures on $\mathbb {R}$ solving the kinetic-type evolution equation $\partial _{t} \mu _{t} + \mu _{t} = Q(\mu _{t})$ where $Q$ is a smoothing transformation on $\mathbb {R}$. This problem has been investigated earlier, e.g. by Bassetti and Ladelli [Ann. Appl. Probab. 22(5): 1928–1961, 2012] and Bogus, Buraczewski and Marynych [Stochastic Process. Appl. 130(2):677–693, 2020]. Combining the refined analysis of the latter paper, which provides a probabilistic description of the solution $\mu _{t}$ as the law of a suitable random sum related to a continuous-time branching random walk at time $t$, with recent advances in the analysis of the extremal positions in the branching random walk we are able to solve the case that has been left open until now. In the course of our work, we significantly weaken the assumptions in the literature that guarantee the existence (and uniqueness) of a solution to the evolution equation $\partial _{t} \mu _{t} + \mu _{t} = Q(\mu _{t})$.

The concept of typical and weighted typical spherical faces for tessellations of the $d$-dimensional unit sphere, generated by $n$ independent random great hyperspheres distributed according to a non-degenerate directional distribution, is introduced and studied. Probabilistic interpretations for such spherical faces are given and their directional distributions are determined. Explicit formulas for the expected $f$-vector, the expected spherical Quermaßintegrals and the expected spherical intrinsic volumes are found in the isotropic case. Their limiting behaviour as $n\to \infty $ is discussed and compared to the corresponding notions and results in the Euclidean case. The expected statistical dimension and a problem related to intersection probabilities of spherical random polytopes is investigated.

We introduce a Cannings model with directional selection via a paintbox construction and establish a strong duality with the line counting process of a new Cannings ancestral selection graph in discrete time. This duality also yields a formula for the fixation probability of the beneficial type. Haldane’s formula states that for a single selectively advantageous individual in a population of haploid individuals of size $N$ the probability of fixation is asymptotically (as $N\to \infty $) equal to the selective advantage of haploids $s_{N}$ divided by half of the offspring variance. For a class of offspring distributions within Kingman attraction we prove this asymptotics for sequences $s_{N}$ obeying $N^{-1} \ll s_{N} \ll N^{-1/2} $, which is a regime of “moderately weak selection”. It turns out that for $ s_{N} \ll N^{-2/3} $ the Cannings ancestral selection graph is so close to the ancestral selection graph of a Moran model that a suitable coupling argument allows to play the problem back asymptotically to the fixation probability in the Moran model, which can be computed explicitly.

Consider a random vector $(V_{1}, \dots , V_{n})$ where $\{V_{k}\}_{k=1, \dots , n}$ are the first $n$ components of a two-parameter Poisson-Dirichlet distribution $PD(\alpha , \theta )$. In this paper, we derive a decomposition for the components of the random vector, and propose an exact simulation algorithm to sample from the random vector. Moreover, a special case arises when $\theta /\alpha $ is a positive integer, for which we present a very fast modified simulation algorithm using a compound geometric representation of the decomposition. Numerical examples are provided to illustrate the accuracy and effectiveness of our algorithms.

Random walks on dynamic graphs have received increasingly more attention from different academic communities over the last decade. Despite the relatively large literature, little is known about random walks that construct the graph where they walk while moving around. In this paper we study one of the simplest conceivable discrete time models of this kind, which works as follows: before every walker step, with probability $p$ a new leaf is added to the vertex currently occupied by the walker. The model grows trees and we call it the Bernoulli Growth Random Walk (BGRW). We show that the BGRW walker is transient and has a well-defined linear speed $c(p)>0$ for any $0<p\leq 1$. Moreover, we show that the tree as seen by the walker converges (in a suitable sense) to a random tree that is one-ended. Some natural open problems about this tree and variants of our model are collected at the end of the paper.

We give a simple proof of a central limit theorem for linear statistics of the circular $\beta $-ensembles which is valid at almost microscopic scales for functions of class $C^{3}$. Using a coupling introduced by Valkò and Viràg [48], we deduce a central limit theorem for the Sine$_{\beta }$ processes. We also discuss connections between our result and the theory of Gaussian Multiplicative Chaos. Based on the results of [37], we show that the exponential of the logarithm of the real (and imaginary) part of the characteristic polynomial of the circular $\beta $-ensembles, regularized at a small mesoscopic scale and renormalized, converges to GMC measures in the subcritical regime. This establishes that the leading order behavior for the extreme values of the logarithm of the characteristic polynomial is consistent with the predictions coming from log-correlated Gaussian field theory.

Matrix concentration inequalities provide information about the probability that a random matrix is close to its expectation with respect to the $\ell _{2}$ operator norm. This paper uses semigroup methods to derive sharp nonlinear matrix inequalities. The main result is that the classical Bakry–Émery curvature criterion implies subgaussian concentration for “matrix Lipschitz” functions. This argument circumvents the need to develop a matrix version of the log-Sobolev inequality, a technical obstacle that has blocked previous attempts to derive matrix concentration inequalities in this setting. The approach unifies and extends much of the previous work on matrix concentration. When applied to a product measure, the theory reproduces the matrix Efron–Stein inequalities due to Paulin et al. It also handles matrix-valued functions on a Riemannian manifold with uniformly positive Ricci curvature.

We study the two-dimensional stochastic nonlinear wave equation (SNLW) and stochastic nonlinear heat equation (SNLH) with a quadratic nonlinearity, forced by a fractional derivative (of order $\alpha > 0$) of a space-time white noise. In particular, we show that the well-posedness theory breaks at $\alpha = \frac 12$ for SNLW and at $\alpha = 1$ for SNLH. This provides a first example showing that SNLW behaves less favorably than SNLH. (i) As for SNLW, Deya (2020) essentially proved its local well-posedness for $0 < \alpha < \frac 12$. We first revisit this argument and establish multilinear smoothing of order $\frac 14$ on the second order stochastic term in the spirit of a recent work by Gubinelli, Koch, and Oh (2018). This allows us to simplify the local well-posedness argument for some range of $\alpha $. On the other hand, when $\alpha \geq \frac 12$, we show that SNLW is ill-posed in the sense that the second order stochastic term is not a continuous function of time with values in spatial distributions. This shows that a standard method such as the Da Prato-Debussche trick or its variant, based on a higher order expansion, breaks down for $\alpha \ge \frac 12$. (ii) As for SNLH, we establish analogous results with a threshold given by $\alpha = 1$.

These examples show that in the case of rough noises, the existing well-posedness theory for singular stochastic PDEs breaks down before reaching the critical values ($\alpha = \frac 34$ in the wave case and $\alpha = 2$ in the heat case) predicted by the scaling analysis (due to Deng, Nahmod, and Yue (2019) in the wave case and due to Hairer (2014) in the heat case).

The Poland-Scheraga model is a celebrated model for the denaturation transition of DNA, which has been widely used in the bio-physical literature to study, and investigated by mathematicians. In the original model, only opposite bases of the two strands can be paired together, but a generalized version of this model has recently been introduced, and allows for mismatches in the pairing of the two strands, and for different strand lengths. This generalized Poland-Scheraga (gPS) model has only been studied recently in the case of homogeneous interactions, then with disordered interactions perturbed by an i.i.d. field. The present paper considers a disordered version of the gPS model which is more appropriate to depict the inhomogeneous composition of the two strands (in particular interactions are perturbed in a strongly dependent manner): we study the question of the influence of disorder on the denaturation transition, and our main results provide criteria for disorder (ir)-relevance, both in terms of critical points and of order of the phase transition. Surprisingly, we find that criteria for disorder relevance depend on the law of the disorder field. We discuss this with regards to Harris’ prediction for disordered systems.

We derive moment and tail estimates for Gaussian chaoses of arbitrary order with values in Banach spaces. We formulate a conjecture regarding two-sided estimates and show that it holds in a certain class of Banach spaces including $L_{q}$ spaces. As a corollary we obtain two-sided bounds for moments of chaoses with values in $L_{q}$ spaces based on exponential random variables.

Consider two laws $P$ and $Q$ of multidimensional possibly explosive diffusions with common diffusion coefficient $\mathfrak {a}$ and drift coefficients $\mathfrak {b}$ and $\mathfrak {b}+ \mathfrak {a}\mathfrak {c}$, respectively, and the law $P^{\circ }$ of an auxiliary diffusion with diffusion coefficient $\langle \mathfrak {c}, \mathfrak {a}\mathfrak {c}\rangle ^{-1}\mathfrak {a}$ and drift coefficient $\langle \mathfrak {c}, \mathfrak {a}\mathfrak {c}\rangle ^{-1}\mathfrak {b}$. We show that $P \ll Q$ if and only if the auxiliary diffusion $P^{\circ }$ explodes almost surely and that $P\perp Q$ if and only if the auxiliary diffusion $P^{\circ }$ almost surely does not explode. As applications we derive a Khasminskii-type integral test for absolute continuity and singularity, an integral test for explosion of time-changed Brownian motion, and we discuss applications to mathematical finance.

In this paper we consider the clustering coefficient, and clustering function in a random graph model proposed by Krioukov et al. in 2010. In this model, nodes are chosen randomly inside a disk in the hyperbolic plane and two nodes are connected if they are at most at a certain hyperbolic distance from each other. It has been previously shown that this model has various properties associated with complex networks, including a power-law degree distribution, “short distances” and a non-vanishing clustering coefficient. The model is specified using three parameters: the number of nodes $n$, which we think of as going to infinity, and $\alpha , \nu > 0$, which we think of as constant. Roughly speaking, the parameter $\alpha $ controls the power law exponent of the degree sequence and $\nu $ the average degree.

Here we show that the clustering coefficient tends in probability to a constant $\gamma $ that we give explicitly as a closed form expression in terms of $\alpha , \nu $ and certain special functions. This improves earlier work by Gugelmann et al., who proved that the clustering coefficient remains bounded away from zero with high probability, but left open the issue of convergence to a limiting constant. Similarly, we are able to show that $c(k)$, the average clustering coefficient over all vertices of degree exactly $k$, tends in probability to a limit $\gamma (k)$ which we give explicitly as a closed form expression in terms of $\alpha , \nu $ and certain special functions. We are able to extend this last result also to sequences $(k_{n})_{n}$ where $k_{n}$ grows as a function of $n$. Our results show that $\gamma (k)$ scales differently, as $k$ grows, for different ranges of $\alpha $. More precisely, there exists constants $c_{\alpha ,\nu }$ depending on $\alpha $ and $\nu $, such that as $k \to \infty $, $\gamma (k) \sim c_{\alpha ,\nu } \cdot k^{2 - 4\alpha }$ if $\frac {1}{2} < \alpha < \frac {3}{4}$, $\gamma (k) \sim c_{\alpha ,\nu } \cdot \log (k) \cdot k^{-1}$ if $\alpha =\frac {3}{4}$ and $\gamma (k) \sim c_{\alpha ,\nu } \cdot k^{-1}$ when $\alpha > \frac {3}{4}$. These results contradict a claim of Krioukov et al., which stated that $\gamma (k)$ should always scale with $k^{-1}$ as we let $k$ grow.

We obtain an integral formula for the distribution of the first hitting time of the origin for one-dimensional α-stable processes , where . We also find a spectral-type integral formula for the transition operators of killed upon hitting the origin. Both expressions involve exponentially growing oscillating functions, which play a role of generalised eigenfunctions for .

By using the heat kernel parameter expansion with respect to the frozen SDEs, the intrinsic derivative is estimated for the law of Mckean-Vlasov SDEs with respect to the initial distribution. As an application, the total variation distance between the laws of two solutions is bounded by the Wasserstein distance for initial distributions. These extend some recent results proved for distribution-free noise by using the coupling method and Malliavin calculus.

We study the eigenvectors and eigenvalues of random matrices with iid entries. Let N be a random matrix with iid entries which have symmetric distribution. For each unit eigenvector v of N our main results provide a small ball probability bound for linear combinations of the coordinates of v. Our results generalize the works of Meehan and Nguyen [59] as well as Touri and the second author [67, 68, 69] for random symmetric matrices. Along the way, we provide an optimal estimate of the probability that an iid matrix has simple spectrum, improving a recent result of Ge [37]. Our techniques also allow us to establish analogous results for the adjacency matrix of a random directed graph, and as an application we establish controllability properties of network control systems on directed graphs.

This paper is concerned with the existence of multiple points of Gaussian random fields. Under the framework of Dalang et al. (2017), we prove that, for a wide class of Gaussian random fields, multiple points do not exist in critical dimensions. The result is applicable to fractional Brownian sheets and the solutions of systems of stochastic heat and wave equations.

We prove that properly rescaled large planar Eulerian triangulations converge to the Brownian map. This result requires more than a standard application of the methods that have been used to obtain the convergence of other families of planar maps to the Brownian map, as the natural distance for Eulerian triangulations is a canonical oriented pseudo-distance. To circumvent this difficulty, we adapt the layer decomposition method, as formalized by Curien and Le Gall in [13], which yields asymptotic proportionality between three natural distances on planar Eulerian triangulations: the usual graph distance, the canonical oriented pseudo-distance, and the Riemannian metric. This notably gives the first mathematical proof of a convergence to the Brownian map for maps endowed with their Riemannian metric. Along the way, we also construct new models of infinite random maps, as local limits of large planar Eulerian triangulations.

We prove the hydrodynamic limit for a one dimensional harmonic chain with a random flip of the momentum sign. The system is open and subject to two thermostats at the boundaries and to an external tension at one of the endpoints. Under a diffusive scaling of space-time, we prove that the empirical profiles of the two locally conserved quantities, the volume stretch and the energy, converge to the solution of a non-linear diffusive system of conservative partial differential equations.

We study the stochastic system of interacting neurons introduced in [5] and in [10] in a diffusive scaling. The system consists of N neurons, each spiking randomly with rate depending on its membrane potential. At its spiking time, the potential of the spiking neuron is reset to 0 and all other neurons receive an additional amount of potential which is a centred random variable of order $1∕\sqrt{N}.$ In between successive spikes, each neuron’s potential follows a deterministic flow. We prove the convergence of the system, as $N\to \mathrm{\infty }$, to a limit nonlinear jumping stochastic differential equation driven by Poisson random measure and an additional Brownian motion W which is created by the central limit theorem. This Brownian motion is underlying each particle’s motion and induces a common noise factor for all neurons in the limit system. Conditionally on $W,$ the different neurons are independent in the limit system. This is the conditional propagation of chaos property. We prove the well-posedness of the limit equation by adapting the ideas of [12] to our frame. To prove the convergence in distribution of the finite system to the limit system, we introduce a new martingale problem that is well suited for our framework. The uniqueness of the limit is deduced from the exchangeability of the underlying system.

Suppose that X, Y are Hilbert-space-valued continuous-path martingales such that Y is differentially subordinate to X. The paper contains the proof of sharp estimates between p-th moments of Y and the maximal function of X for $0<p<1$. The proof rests on Burkholder’s method and exploits a certain special function of three variables, enjoying appropriate size and concavity requirements. The analysis reveals an unexpected phase transition between the cases $0<p<1∕2$ and $1∕2\le p<1$. The latter case is relatively simple: the special function is essentially quadratic and the best constant is equal to $\sqrt{2∕p}$. The analysis of the former case is much more intricate and involves the study of a non-linear ordinary differential equation.

We study the long time behaviour of a large class of diffusion processes on ${\mathbb{R}}^N$, generated by second order differential operators of (possibly) degenerate type. The operators that we consider need not satisfy the Hörmander Condition (HC). Instead, they satisfy the so-called UFG condition, introduced by Herman, Lobry and Sussman in the context of geometric control theory and later by Kusuoka and Stroock. We demonstrate the importance of the class of UFG processes in several respects: i) we show that UFG processes constitute a family of SDEs which exhibit, in general, multiple invariant measures (i.e. they are in general non-ergodic) and for which one is able to describe a systematic procedure to study the basin of attraction of each invariant measure (equilibrium state). ii) We use an explicit change of coordinates to prove that every UFG diffusion can be, at least locally, represented as a system consisting of an SDE coupled with an ODE, where the ODE evolves independently of the SDE part of the dynamics. iii) As a result, UFG diffusions are inherently “less smooth” than hypoelliptic SDEs; more precisely, we prove that UFG processes do not admit a density with respect to Lebesgue measure on the entire space, but only on suitable time-evolving submanifolds, which we describe. iv) We show that our results and techniques, which we devised for UFG processes, can be applied to the study of the long-time behaviour of non-autonomous hypoelliptic SDEs and therefore produce several results on this latter class of processes as well. v) Because processes that satisfy the (uniform) parabolic HC are UFG processes, this paper contains a wealth of results about the long time behaviour of (uniformly) hypoelliptic processes which are non-ergodic.

Consider N balls initially placed in L bins. At each time step take a ball from each non-empty bin and randomly reassign all the balls into the bins. We call this finite Markov chain General Repeated Balls into Bins process. It is a discrete time conservative interacting particles system with parallel updates. Assuming a quantitative chaotic condition on the reassignment rule we prove a quantitative propagation of chaos for this model. We furthermore study some equilibrium properties of the limiting nonlinear process.

We extend our recent result [22] on the central limit theorem for the linear eigenvalue statistics of non-Hermitian matrices X with independent, identically distributed complex entries to the real symmetry class. We find that the expectation and variance substantially differ from their complex counterparts, reflecting (i) the special spectral symmetry of real matrices onto the real axis; and (ii) the fact that real i.i.d. matrices have many real eigenvalues. Our result generalizes the previously known special cases where either the test function is analytic [49] or the first four moments of the matrix elements match the real Gaussian [59, 44]. The key element of the proof is the analysis of several weakly dependent Dyson Brownian motions (DBMs). The conceptual novelty of the real case compared with [22] is that the correlation structure of the stochastic differentials in each individual DBM is non-trivial, potentially even jeopardising its well-posedness.

We are interested in populations in which the fitness of different genetic types fluctuates in time and space, driven by temporal and spatial fluctuations in the environment. For simplicity, our population is assumed to be composed of just two genetic types. Short bursts of selection acting in opposing directions drive to maintain both types at intermediate frequencies, while the fluctuations due to ‘genetic drift’ work to eliminate variation in the population.

We consider first a population with no spatial structure, modelled by an adaptation of the Lambda (or generalised) Fleming-Viot process, and derive a stochastic differential equation as a scaling limit. This amounts to a limit result for a Lambda-Fleming-Viot process in a rapidly fluctuating random environment. We then extend to a population that is distributed across a spatial continuum, which we model through a modification of the spatial Lambda-Fleming-Viot process with selection. In this setting we show that the scaling limit is a stochastic partial differential equation. As is usual with spatially distributed populations, in dimensions greater than one, the ‘genetic drift’ disappears in the scaling limit, but here we retain some stochasticity due to the fluctuations in the environment, resulting in a stochastic p.d.e. driven by a noise that is white in time but coloured in space.

We discuss the (rather limited) situations under which there is a duality with a system of branching and annihilating particles. We also write down a system of equations that captures the frequency of descendants of particular subsets of the population and use this same idea of ‘tracers’, which we learned from HALLATSCHEK and NELSON (2008, [23]) and DURRETT and FAN (2016, [13]), in numerical experiments with a closely related model based on the classical Moran model.

Let f be a stationary isotropic non-degenerate Gaussian field on ${\mathbb{R}}^2$. Assume that $f=q\ast W$ where $q\in L^2({\mathbb{R}}^2)\cap C^2({\mathbb{R}}^2)$ and W is the $L^2$ white noise on ${\mathbb{R}}^2$. We extend a result by Stephen Muirhead and Hugo Vanneuville by showing that, assuming that $q\ast q$ is pointwise non-negative and has fast enough decay, the set $\{f\ge -\ell \}$ percolates with probability one when $\ell >0$ and with probability zero if $\ell \le 0$. We also prove exponential decay of crossing probabilities and uniqueness of the unbounded cluster. To this end, we study a Gaussian field g defined on the torus and establish a superconcentration formula for the threshold $\text{T}(g)$ which is the minimal value such that $\{g\ge -\text{T}(g)\}$ contains a non-contractible loop. This formula follows from a Gaussian Talagrand type inequality.

Inspired by the works in [2] and [11] we introduce what we call k-th-order fluctuation fields and study their scaling limits. This construction is done in the context of particle systems with the property of orthogonal self-duality. This type of duality provides us with a setting in which we are able to interpret these fields as some type of discrete analogue of powers of the well-known density fluctuation field. We show that the weak limit of the k-th order field satisfies a recursive martingale problem that corresponds to the SPDE associated with the kth-power of a generalized Ornstein-Uhlenbeck process.

In this paper we develop a framework for multivariate functional approximation by a suitable Gaussian process via an exchangeable pairs coupling that satisfies a suitable approximate linear regression property, thereby building on work by Barbour (1990) and Kasprzak (2020). We demonstrate the applicability of our results by applying them to joint subgraph counts in an Erdős-Renyi random graph model on the one hand and to vectors of weighted, degenerate U-processes on the other hand. As a concrete instance of the latter class of examples, we provide a bound for the functional approximation of a vector of success runs of different lengths by a suitable Gaussian process which, even in the situation of just a single run, would be outside the scope of the existing theory.

We consider a stochastic Volterra integral equation with regular path-dependent coefficients and a Brownian motion as integrator in a multidimensional setting. Under an imposed absolute continuity condition, the unique solution is a semimartingale that admits almost surely Hölder continuous paths. Based on functional Itô calculus, we prove that the support of its law in Hölder norms can be described by a flow of mild solutions to ordinary integro-differential equations that are constructed by means of the vertical derivative of the diffusion coefficient.

This work exhibits a novel phase transition for the classical stochastic block model (SBM). In addition we study the SBM in the corresponding near-critical regime, and find the scaling limit for the component sizes. The two-parameter stochastic process arising in the scaling limit, an analogue of the standard Aldous’ multiplicative coalescent, is interesting in its own right. We name it the (standard) Interacting Multiplicative Coalescent. To the best of our knowledge, this object has not yet appeared in the literature.

The paper defines and studies thin times which are random times whose graph is contained in a countable union of graphs of stopping times with respect to a reference filtration $\mathbb{F}$. We show that a generic random time can be decomposed into thin and thick parts, where the second is a random time avoiding all $\mathbb{F}$-stopping times. Then, for a given random time τ, we introduce ${\mathbb{F}}^{\mathrm{\tau }}$, the smallest right-continuous filtration containing $\mathbb{F}$ and making τ a stopping time, and we show that, for a thin time τ, each $\mathbb{F}$-martingale is an ${\mathbb{F}}^{\mathrm{\tau }}$-semimartingale, i.e., the hypothesis $({\mathcal{H}}^{\prime })$ for $(\mathbb{F},{\mathbb{F}}^{\mathrm{\tau }})$ holds. We present applications to honest times, which can be seen as last passage times, showing classes of filtrations which can only support thin honest times, or can accommodate thick honest times as well.

We prove that for $\mathrm{\kappa }\in (0,8)$, if $({\mathrm{\eta }}_1,{\mathrm{\eta }}_2)$ is a 2-SLE${}_{\mathrm{\kappa }}$pair in a simply connected domain D with an analytic boundary point $z_0$, then as $r\to 0^{+}$, $\mathbb{P}[\mathrm{dist}(z_0,{\mathrm{\eta }}_j)<r,j=1,2]$ converges to a positive number for some $\mathrm{\alpha }>0$, which is called the two-curve Green’s function. The exponent α equals $\frac{12}{\mathrm{\kappa }}-1$ or $2(\frac{12}{\mathrm{\kappa }}-1)$ depending on whether $z_0$ is one of the endpoints of ${\mathrm{\eta }}_1$ or ${\mathrm{\eta }}_2$. We also find the convergence rate and the exact formula for the Green’s function up to a multiplicative constant. To derive these results, we construct two-dimensional diffusion processes and use orthogonal polynomials to obtain their transition density.

We study the macroscopic evolution of the growing cluster in the exactly solvable corner growth model with independent exponentially distributed waiting times. The rates of the exponentials are given by an addivitely separable function of the site coordinates. When computing the growth process (last-passage times) at each site, the horizontal and vertical additive components of the rates are allowed to also vary respectively with the column and row number of that site. This setting includes several models of interest from the literature as special cases. Our main result provides simple explicit variational formulas for the a.s. first-order asymptotics of the growth process under a decay condition on the rates. Subject to further mild conditions, we prove the existence of the limit shape and describe it explicitly. We observe that the boundary of the limit shape can develop flat segments adjacent to the axes and spikes along the axes. Furthermore, we record the formation of persistent macroscopic spikes and crevices in the cluster that are nonetheless not visible in the limit shape. As an application of the results for the growth process, we compute the flux function and limiting particle profile for the TASEP with the step initial condition and disorder in the jump rates of particles and holes. Our methodology is based on concentration bounds and estimating the boundary exit probabilities of the geodesics in the increment-stationary version of the model, with the only input from integrable probability being the distributional invariance of the last-passage times under permutations of columns and rows.

We present a large deviation principle at speed N for the largest eigenvalue of some additively deformed Wigner matrices. In particular this includes Gaussian ensembles with full-rank general deformation. For the non-Gaussian ensembles, the deformation should be diagonal, and we assume that the laws of the entries have sharp sub-Gaussian Laplace transforms and satisfy certain concentration properties. For these latter ensembles we establish the large deviation principle in a restricted range $(-\mathrm{\infty },x_c)$, where $x_c$ depends on the deformation only and can be infinite.

We extend a model of positive feedback and contagion in large mean-field systems, by introducing a common source of noise driven by Brownian motion. Although the driving dynamics are continuous, the positive feedback effect can lead to ‘blow-up’ phenomena whereby solutions develop jump-discontinuities. Our main results are twofold and concern the conditional McKean–Vlasov formulation of the model. First and foremost, we show that there are global solutions to this McKean–Vlasov problem, which can be realised as limit points of a motivating particle system with common noise. Furthermore, we derive results on the occurrence of blow-ups, thereby showing how these events can be triggered or prevented by the pathwise realisations of the common noise.

We consider a broad class of Approximate Message Passing (AMP) algorithms defined as a Lipschitzian functional iteration in terms of an $n\times n$ random symmetric matrix A. We establish universality in noise for this AMP in the n-limit and validate this behavior in a number of AMPs popularly adapted in compressed sensing, statistical inferences, and optimizations in spin glasses.

We consider metric graph Gaussian free field (GFF) defined on polygons of $\mathrm{\delta }{\mathbb{Z}}^2$ with alternating boundary data. The crossing probabilities for level-set percolation of metric graph GFF have scaling limits. When the boundary data is well-chosen, the scaling limits of crossing probabilities can be explicitly constructed as “fusion” of multiple SLE${}_4$ pure partition functions.

We study several matrix diffusion processes constructed from a unitary Brownian motion. In particular, we use the Stiefel fibration to lift the Brownian motion of the complex Grassmannian to the complex Stiefel manifold and deduce a skew-product decomposition of the Stiefel Brownian motion. As an application, we prove asymptotic laws for the determinants of the block entries of the unitary Brownian motion.

The soft and hard edge scaling limits of β-ensembles can be characterized as the spectra of certain random Sturm-Liouville operators [12, 15]. It has been shown that by tuning the parameter of the hard edge process one can obtain the soft edge process as a scaling limit [3, 12, 14]. We prove that this limit can be realized on the level of the corresponding random operators. More precisely, the random operators can be coupled in a way so that the scaled versions of the hard edge operators converge to the soft edge operator a.s. in the norm resolvent sense.

We consider the least singular value of $\mathit{M}={\mathit{R}}^{\ast }\mathit{X}\mathit{T}+{\mathit{U}}^{\ast }\mathit{Y}\mathit{V}$, where R, T, U, V are independent Haar-distributed unitary matrices and X, Y are deterministic diagonal matrices. Under weak conditions on X and Y, we show that the limiting distribution of the least singular value of M, suitably rescaled, is the same as the limiting distribution for the least singular value of a matrix of i.i.d. Gaussian random variables. Our proof is based on the dynamical method used by Che and Landon to study the local spectral statistics of sums of Hermitian matrices.

We obtain upper bounds for the rates of convergence for the simple random walk Green’s function in the domains $D_{\mathit{\alpha }}=D_{\mathit{\alpha }}(n)=\{r{e}^{i\mathit{𝜃}}\in \mathbb{C}:0<\mathit{𝜃}<2\mathit{\pi }-\mathit{\alpha },0<r<2n\}-z_0$, where $z_0\in {\mathbb{Z}}^2$ is a point closest to $n{e}^{i(\mathit{\pi }-\mathit{\alpha }∕2)}$. The rate depends on the angle of the wedge and is what was suggested by the sharpest available results in the extreme cases $\mathit{\alpha }=0$ and $\mathit{\alpha }=\mathit{\pi }$. Our proof uses the KMT coupling between random walk and Brownian motion.

Let G be an infinite connected graph with vertex set V. Let $\{S_n:n\in {\mathbb{N}}_0\}$ be the simple random walk on G and let $\{\mathit{\xi }(v):v\in V\}$ be a collection of i.i.d. random variables which are independent of the random walk. Define the random walk in random scenery as $T_n={\sum }_{k=0}^n\mathit{\xi }(S_k)$, and the normalization variables $V_n={({\sum }_{k=0}^n{\mathit{\xi }}^2(S_k))}^{1∕2}$ and $L_{n,2}={({\sum }_{v\in V}{\ell }_n^2(v))}^{1∕2}$. For $G={\mathbb{Z}}^d$ and $G={\mathbb{T}}_d$, the d-ary tree, we provide large deviations results for the self-normalized process $T_n\sqrt{n}∕(L_{n,2}{V}_n)$ under only finite moment assumptions on the scenery.

In the present paper, we determine the full spectrum of the simple random walk on finite, complete d-ary trees. We also find an eigenbasis for the transition matrix. As an application, we apply our results to get a lower bound for the interchange process on complete, finite d-ary trees, which we conjecture to be sharp.

We show that there is “no stable free field of index $\mathit{\alpha }\in (1,2)$”, in the following sense. It was proved in [4] that subject to a fourth moment assumption, any random generalised function on a domain D of the plane, satisfying conformal invariance and a natural domain Markov property, must be a constant multiple of the Gaussian free field. In this article we show that the existence of $(1+\mathrm{𝜀})$ moments is sufficient for the same conclusion. A key idea is a new way of exploring the field, where (instead of looking at the more standard circle averages) we start from the boundary and discover averages of the field with respect to a certain “hitting density” of Itô excursions.

We study the relationship between functional inequalities for a Markov kernel on a metric space X and inequalities of transportation distances on the space of probability measures $\mathcal{P}(X)$. Extending results of Luise and Savaré on Hellinger–Kantorovich contraction inequalities for the particular case of the heat semigroup on an $RCD(K,\mathrm{\infty })$ metric space, we show that more generally, such contraction inequalities are equivalent to reverse Poincaré inequalities. We also adapt the “dynamic dual” formulation of the Hellinger–Kantorovich distance to define a new family of divergences on $\mathcal{P}(X)$ which generalize the Rényi divergence, and we show that contraction inequalities for these divergences are equivalent to the reverse logarithmic Sobolev and Wang Harnack inequalities. We discuss applications including results on the convergence of Markov processes to equilibrium, and on quasi-invariance of heat kernel measures in finite and infinite-dimensional groups.

We examine a two-dimensional Coulomb gas consisting of n identical repelling point charges at an arbitrary inverse temperature β, subjected to a suitable external field.

We prove that the gas is effectively localized to a small neighbourhood of the droplet – the support of the equilibrium measure determined by the external field. More precisely, we prove that the distance between the droplet and the vacuum is with very high probability at most proportional to

$$\sqrt{\frac{logn}{\mathit{\beta }n}}.$$

This order of magnitude is known to be “tight” when $\mathit{\beta }=1$ and the external field is radially symmetric.

In addition, we prove estimates for the one-point function in a neighbourhood of the droplet, proving in particular a fast uniform decay as one moves beyond a distance roughly of the order $\sqrt{\frac{logn}{\mathit{\beta }n}}$ from the droplet.

We analyse the metastable behaviour of the dilute Curie–Weiss model subject to a Glauber dynamics. The model is a random version of a mean-field Ising model, where the coupling coefficients are Bernoulli random variables with mean $p\in (0,1)$. This model can be also viewed as an Ising model on the Erdős–Rényi random graph with edge probability p. The system is a Markov chain where spins flip according to a Metropolis dynamics at inverse temperature β. We compute the average time the system takes to reach the stable phase when it starts from a certain probability distribution on the metastable state (called the last-exit biased distribution), in the regime where $N\to \mathrm{\infty }$, $\mathit{\beta }>{\mathit{\beta }}_c=1$ and h is positive and small enough. We obtain asymptotic bounds on the probability of the event that the mean metastable hitting time is approximated by that of the Curie–Weiss model. The proof uses the potential theoretic approach to metastability and concentration of measure inequalities.

We derive multi-level concentration inequalities for polynomials in independent random variables with an α-sub-exponential tail decay. A particularly interesting case is given by quadratic forms $f(X_1,\dots ,X_n)=⟨X,AX⟩$, for which we prove Hanson–Wright-type inequalities with explicit dependence on various norms of the matrix A. A consequence of these inequalities is a two-level concentration inequality for quadratic forms in α-sub-exponential random variables, such as quadratic Poisson chaos.

We provide various applications of these inequalities. Among them are generalizations of some results proven by Rudelson and Vershynin from sub-Gaussian to α-sub-exponential random variables, i. e. concentration of the Euclidean norm of the linear image of a random vector and concentration inequalities for the distance between a random vector and a fixed subspace.

This paper develops a theory for completely random measures in the framework of free probability. A general existence result for free completely random measures is established, and in analogy to the classical work of Kingman it is proved that such random measures can be decomposed into the sum of a purely atomic part and a (freely) infinitely divisible part. The latter part (termed a free Lévy basis) is studied in detail in terms of the free Lévy-Khintchine representation and a theory parallel to the classical work of Rajput and Rosiński is developed. Finally a Lévy-Itô type decomposition for general free Lévy bases is established.

A universal method for the concatenation of a sequence of Markov right processes is established. It is then applied to the continued pasting of two Markov right processes, which can be used for pathwise constructions of locally defined processes like Brownian motions on compact intervals.

We consider exclusion processes with two types of particles which compete strongly with each other. In particular, we focus on the case where one species does not diffuse at all and killing rates of two species are given by monomials with distinct exponents. We study limiting behavior of interfaces which appear by such a strong competition. Consequently, three kinds of limiting behavior of interfaces (vanishing, moving and immovable interfaces as in [9]) are derived directly from our interacting particle system taking advantage of hydrodynamic limit procedure with singular limit for annihilation dynamics.

In the setting of reversible continuous-time Markov chains, the $C{D}_{\mathrm{{\rm Y}}}$ condition has been shown recently to be a consistent analogue to the Bakry-Émery condition in the diffusive setting in terms of proving Li-Yau inequalities under a finite dimension term and proving the modified logarithmic Sobolev inequality under a positive curvature bound. In this article we examine the case where both is given, a finite dimension term and a positive curvature bound. For this purpose we introduce the $C{D}_{\mathrm{{\rm Y}}}(\mathrm{\kappa },F)$ condition, where the dimension term is expressed by a so called $CD$-function F. We derive functional inequalities relating the entropy to the Fisher information, which we will call entropy-information inequalities. Further, we deduce applications of entropy-information inequalities such as ultracontractivity bounds, exponential integrability of Lipschitz functions, finite diameter bounds and a modified version of the celebrated Nash inequality.

Consider an elliptic parameter k; we introduce a family of $Z^u$-Dirac operators ${(\mathsf{K}(u))}_{u\in \mathcal{C}}$, relate them to the Z-massive Laplacian of [11], and extend to the full Z-invariant case the results of Kenyon [45] on discrete holomorphic and harmonic functions, which correspond to the case $k=0$. We prove through combinatorial identities, how and why the $Z^u$-Dirac and Z-massive Laplacian operators appear in the Z-invariant Ising model, considering the case of infinite and finite isoradial graphs. More precisely, consider the dimer model on the Fisher graph ${\mathsf{G}}^{\mathrm{F}}$ arising from a Z-invariant Ising model. We express coefficients of the inverse Fisher Kasteleyn operator as a function of the inverse $Z^u$-Dirac operator and also as a function of the Z-massive Green function; in particular this proves a (massive) random walk representation of important observables of the Ising model. We prove that the squared partition function of the Ising model is equal, up to a constant, to the determinant of the Z-massive Laplacian operator with specific boundary conditions, the latter being the partition function of rooted spanning forests. To show these results, we relate the inverse Fisher Kasteleyn operator and that of the dimer model on the bipartite graph ${\mathsf{G}}^{\mathrm{Q}}$ arising from the XOR-Ising model, and we prove matrix identities between the Kasteleyn matrix of ${\mathsf{G}}^{\mathrm{Q}}$ and the $Z^u$-Dirac operator, that allow to reach inverse matrices as well as determinants.

Following the strategy proposed by Makarov and Smirnov [24] in 2009 (see also [3, 2] for theoretical physics arguments), we provide technical details for the proof of convergence of massive loop-erased random walks to the chordal mSLE(2) process. As no follow-up to [24] appeared since then, we believe that such a treatment might be of interest to the community. We do not require any regularity of the limiting planar domain near its degenerate prime ends a and b except that $({\mathrm{\Omega }}^{\mathit{\delta }},a^{\mathit{\delta }},b^{\mathit{\delta }})$ are assumed to be ‘close discrete approximations’ to $(\mathrm{\Omega },a,b)$ near a and b in the sense of [13].

This paper presents new limit theorems for power variations of fractional type symmetric infinitely divisible random fields. More specifically, the random field $X={(X(t))}_{t\in {[0,1]}^d}$ is defined as an integral of a kernel function g with respect to a symmetric infinitely divisible random measure L and is observed on a grid with mesh size $n^{-1}$. As $n\to \mathrm{\infty }$, the first order limits are obtained for power variation statistics constructed from rectangular increments of X. The present work is mostly related to [8, 9], who studied a similar problem in the case $d=1$. We will see, however, that the asymptotic theory in the random field setting is much richer compared to [8, 9] as it contains new limits, which depend on the precise structure of the kernel g. We will give some important examples including the Lévy moving average field, the well-balanced symmetric linear fractional β-stable sheet, and the moving average fractional β-stable field, and discuss potential consequences for statistical inference.

SupOU processes are superpositions of Ornstein-Uhlenbeck type processes with a random intensity parameter. They are stationary processes whose marginal distribution and dependence structure can be specified independently. Integrated supOU processes have then stationary increments and satisfy central and non-central limit theorems. Their moments, however, can display an unusual behavior known as “intermittency”. We show here that intermittency can also appear when the processes have a heavy tailed marginal distribution and, in particular, an infinite variance.

We study a random walk on a point process given by an ordered array of points $({\mathit{\omega }}_k,k\in \mathbb{Z})$ on the real line. The distances ${\mathit{\omega }}_{k+1}-{\mathit{\omega }}_k$ are i.i.d. random variables in the domain of attraction of a β-stable law, with $\mathit{\beta }\in (0,1)\cup (1,2)$. The random walk has i.i.d. jumps such that the transition probabilities between ${\mathit{\omega }}_k$ and ${\mathit{\omega }}_{\ell }$ depend on $\ell -k$ and are given by the distribution of a $\mathbb{Z}$-valued random variable in the domain of attraction of an α-stable law, with $\mathit{\alpha }\in (0,1)\cup (1,2)$. Since the defining variables, for both the random walk and the point process, are heavy-tailed, we speak of a Lévy flight on a Lévy random medium. For all combinations of the parameters α and β, we prove the annealed functional limit theorem for the suitably rescaled process, relative to the optimal Skorokhod topology in each case. When the limit process is not càdlàg, we prove convergence of the finite-dimensional distributions. When the limit process is deterministic, we also prove a limit theorem for the fluctuations, again relative to the optimal Skorokhod topology.

We study the distribution of the area under the normalized excursion of a spectrally positive stable Lévy process L, as well as the area under its meander, and under L conditioned to stay positive. Our results involve a special case of Wright’s function, which may be seen as a generalization of the classic Airy function appearing in similar Brownian’s areas.

We study the rate of convergence of the empirical spectral distribution of products of independent non-Hermitian random matrices to the power of the Circular Law. The distance to the deterministic limit distribution will be measured in terms of a uniform Kolmogorov-like distance. First, we prove that for products of Ginibre matrices, the optimal rate is given by $\mathcal{O}(1∕\sqrt{n})$, which is attained with overwhelming probability up to a logarithmic correction. Avoiding the edge, the rate of convergence of the mean empirical spectral distribution is even faster. Second, we show that also products of matrices with independent entries attain this optimal rate in the bulk up to a logarithmic factor. In the case of Ginibre matrices, we apply a saddlepoint approximation to a double contour integral representation of the density and in the case of matrices with independent entries we make use of techniques from local laws.

For two independent Lévy processes ξ and η and an exponentially distributed random variable τ with parameter $q>0$ that is independent of ξ and η, the killed exponential functional is given by $V_{q,\mathrm{\xi },\mathrm{\eta }}:={\int }_0^{\mathit{\tau }}{\mathrm{e}}^{-{\mathrm{\xi }}_{s-}}\mathrm{d}{\mathrm{\eta }}_s$. With the killed exponential functional arising as the stationary distribution of a Markov process, we calculate the infinitesimal generator of the process and use it to derive different distributional equations describing the law of $V_{q,\mathrm{\xi },\mathrm{\eta }}$, as well as functional equations for its Lebesgue density in the absolutely continuous case. Various special cases and examples are considered, yielding more explicit information on the law of the killed exponential functional and illustrating the applications of the equations obtained. Interpreting the case $q=0$ as $\mathit{\tau }=\mathrm{\infty }$ leads to the classical exponential functional ${\int }_0^{\mathrm{\infty }}{\mathrm{e}}^{-{\mathrm{\xi }}_{s-}}\mathrm{d}{\mathrm{\eta }}_s$, allowing to extend many previous results to include killing.

We consider a two-type reducible branching Brownian motion, defined as a particle system on the real line in which particles of two types move according to independent Brownian motions and create offspring at a constant rate. Particles of type 1 can give birth to particles of types 1 and 2, but particles of type 2 only give birth to descendants of type 2. Under some specific conditions, Biggins [10] shows that this process exhibits an anomalous spreading behaviour: the rightmost particle at time t is much further than the expected position for the rightmost particle in a branching Brownian motion consisting only of particles of type 1 or of type 2. This anomalous spreading also has been investigated from a reaction-diffusion equation standpoint by Holzer [26, 27]. The aim of this article is to study the asymptotic behaviour of the position of the furthest particles in the two-type reducible branching Brownian motion, obtaining in particular tight estimates for the median of the maximal displacement.

We show that the partition function of the multi-layer semi-discrete directed polymer converges in the intermediate disorder regime to the partition function for the multi-layer continuum polymer introduced by O’Connell and Warren in [24]. This verifies, modulo a previously hidden constant, an outstanding conjecture proposed by Corwin and Hammond [5]. A consequence is the identification of the KPZ line ensemble as logarithms of ratios of consecutive layers of the continuum partition function. Other properties of the continuum partition function, such as continuity, strict positivity and contour integral formulas to compute mixed moments, are also identified from this convergence result.

We give a simple proof for recurrence of vertex reinforced jump process on ${\mathbb{Z}}^d$, under strong reinforcement. Moreover, we show how the previous result implies that linearly edge-reinforced random walk on ${\mathbb{Z}}^d$ is recurrent for strong reinforcement. Finally, we prove that the $H^{(2|2)}$ model on ${\mathbb{Z}}^d$ localizes at strong disorder. Even though these results are well-known, we propose a unified approach, which also has the advantage to provide shorter proofs, and relies on estimating fractional moments, introduced by Aizenman and Molchanov.

We consider a class of Backward Stochastic Differential Equations with superlinear driver process f adapted to a filtration supporting at least a d dimensional Brownian motion and a Poisson random measure on ${\mathbb{R}}^m\setminus \{0\}$. We consider the following class of terminal conditions: ${\mathrm{\xi }}_1=\mathrm{\infty }\cdot {\mathbf{1}}_{\{{\mathit{\tau }}_1\le T\}}$ where ${\mathit{\tau }}_1$ is any stopping time with a bounded density in a neighborhood of T and ${\mathrm{\xi }}_2=\mathrm{\infty }\cdot {\mathbf{1}}_{A_T}$ where $A_t$, $t\in [0,T]$ is a decreasing sequence of events adapted to the filtration ${\mathcal{F}}_t$ that is continuous in probability at T (equivalently, $A_T=\{{\mathit{\tau }}_2>T\}$ where ${\mathit{\tau }}_2$ is any stopping time such that $\mathbb{P}({\mathit{\tau }}_2=T)=0$). In this setting we prove that the minimal supersolutions of the BSDE are in fact solutions, i.e., they attain almost surely their terminal values. We note that the first exit time from a time varying domain of a d-dimensional diffusion process driven by the Brownian motion with strongly elliptic covariance matrix does have a continuous density. Therefore such exit times can be used as ${\mathit{\tau }}_1$ and ${\mathit{\tau }}_2$ to define the terminal conditions ${\mathrm{\xi }}_1$ and ${\mathrm{\xi }}_2$. The proof of existence of the density is based on the classical Green’s functions for the associated PDE.

In this paper, we study the fluctuations of observables of metric measure spaces which are random discrete approximations ${\mathcal{X}}_n$ of a fixed arbitrary (complete, separable) metric measure space $\mathcal{X}=(\mathcal{X},d,\mathrm{\mu })$. These observables $\mathrm{\Phi }({\mathcal{X}}_n)$ are polynomials in the sense of Greven–Pfaffelhuber–Winter, and we show that for a generic model space $\mathcal{X}$, they yield asymptotically normal random variables. However, if $\mathcal{X}$ is a compact homogeneous space, then the fluctuations of the observables are much smaller, and after an adequate rescaling, they converge towards probability distributions which are not Gaussian. Conversely, we prove that if all the fluctuations of the observables $\mathrm{\Phi }({\mathcal{X}}_n)$ are smaller than in the generic case, then the measure metric space $\mathcal{X}$ is compact homogeneous. The proofs of these results rely on the Gromov reconstruction principle, and on an adaptation of the method of cumulants and mod-Gaussian convergence developed by Féray–Méliot–Nikeghbali. As an application of our results, we construct a statistical test of the hypothesis of symmetry of a compact Riemannian manifold.

We show that one-dimensional random walks in a quasi-periodic environment with Liouville frequency generically have an erratic statistical behavior. In the recurrent case we show that neither quenched nor annealed limit theorems hold and both drift and variance exhibit wild oscillations, being logarithmic at some times and almost linear at other times. In the transient case we show that the annealed Central Limit Theorem fails generically. These results are in stark contrast with the Diophantine case where the Central Limit Theorem with linear drift and variance was established by Sinai.

It has recently been shown that complete Bernstein functions of the Laplace operator map the Dirichlet boundary condition of a related elliptic PDE to the Neumann boundary condition. The importance of this mapping consists in being able to convert problems involving non-local operators, like fractional Laplacians, into ones that only involve differential operators. We generalise this result to diffusion operators associated with stochastic differential equations, using a method which is entirely based on stochastic analysis.

Let $f:\mathbb{R}\to \mathbb{R}$ be a stationary centered Gaussian process. For any $R>0$, let ${\mathrm{\nu }}_R$ denote the counting measure of $\{x\in \mathbb{R}\mid f(Rx)=0\}$. Under suitable assumptions on the regularity of f and the decay of its correlation function at infinity, we derive the asymptotics as $R\to +\mathrm{\infty }$ of the central moments of the linear statistics of ${\mathrm{\nu }}_R$. In particular, we derive an asymptotics of order $R^{\frac{p}{2}}$ for the p-th central moment of the number of zeros of f in $[0,R]$. As an application, we prove a functional Law of Large Numbers and a functional Central Limit Theorem for the random measures ${\mathrm{\nu }}_R$. More precisely, after a proper rescaling, ${\mathrm{\nu }}_R$ converges almost surely towards the Lebesgue measure in weak-∗ sense. Moreover, the fluctuation of ${\mathrm{\nu }}_R$ around its mean converges in distribution towards the standard Gaussian White Noise. The proof of our moments estimates relies on a careful study of the k-point function of the zero point process of f, for any $k\ge 2$. Our analysis yields two results of independent interest. First, we derive an equivalent of this k-point function near any point of the large diagonal in ${\mathbb{R}}^k$, thus quantifying the short-range repulsion between zeros of f. Second, we prove a clustering property which quantifies the long-range decorrelation between zeros of f.

Let ${(X_k)}_{k\ge 1}$ and ${(Y_k)}_{k\ge 1}$ be two independent sequences of i.i.d. random variables, with values in a finite and totally ordered alphabet ${\mathcal{A}}_m:=\{1,\dots ,m\}$, $m\ge 2$, having respective probability mass function $p_1^X,\dots ,p_m^X$ and $p_1^Y,\dots ,p_m^Y$. Let $LC{I}_n$ be the length of the longest common and weakly increasing subsequences in $X_1,\mathrm{...},X_n$ and $Y_1,\mathrm{...},Y_n$. Once properly centered and normalized, $LC{I}_n$ is shown to have a limiting distribution which is expressed as a functional of two independent multidimensional Brownian motions.

We consider a bivariate Markov chain $Z={\{Z_k\}}_{k\ge 1}={\{(X_k,Y_k)\}}_{k\ge 1}$ taking values on product space $\mathcal{Z}=\mathcal{X}\times \mathcal{Y}$, where $\mathcal{X}$ is possibly uncountable space and $\mathcal{Y}=\{1,\dots ,|\mathcal{Y}|\}$ is a finite state-space. The purpose of the paper is to find sufficient conditions that guarantee the exponential convergence of smoothing, filtering and predictive probabilities:

$$\underset{n\ge t}{sup}\Vert P(Y_{t:\mathrm{\infty }}\in \cdot |X_{l:n})-P(Y_{t:\mathrm{\infty }}\in \cdot |X_{s:n}){\Vert }_{\mathrm{TV}}\le K_s{\mathit{\alpha }}^t,\text{a.s.}$$

Here $t\ge s\ge l\ge 1$, $K_s$ is $\mathit{\sigma }(X_{s:\mathrm{\infty }})$-measurable finite random variable and $\mathit{\alpha }\in (0,1)$ is fixed. In the second part of the paper, we establish two-sided versions of the above-mentioned convergence. We show that the desired convergences hold under fairly general conditions. A special case of above-mentioned very general model is popular hidden Markov model (HMM). We prove that in HMM-case, our assumptions are more general than all similar mixing-type of conditions encountered in practice, yet relatively easy to verify.

In this paper, existence and uniqueness are proved for path-dependent McKean-Vlasov type SDEs with integrability conditions. Gradient estimates and the Harnack type inequalities are derived in the case that the drifts are Dini continuous in the space variable. These generalize the corresponding results derived for classical functional SDEs with singular coefficients.

An infinite system of point particles placed in ${ℝ}^d$ is studied. Its constituents perform random jumps (walks) with mutual repulsion described by a translation-invariant jump kernel and interaction potential, respectively. The pure states of the system are locally finite subsets of ${ℝ}^d$, which can also be interpreted as locally finite Radon measures. The set of all such measures Γ is equipped with the vague topology and the corresponding Borel σ-field. For a special class ${\mathcal{P}}_{\exp }$ of (sub-Poissonian) probability measures on Γ, we prove the existence of a unique family $\{P_{t,\mathrm{\mu }}:t\ge 0,\mathrm{\mu }\in {\mathcal{P}}_{\exp }\}$ of probability measures on the space of cadlag paths with values in Γ that solves a restricted initial-value martingale problem for the mentioned system. Thereby, a Markov process with cadlag paths is specified which describes the stochastic dynamics of this particle system.

We consider the filtering of continuous-time finite-state hidden Markov models, where the rate and observation matrices depend on unknown time-dependent parameters, for which no prior or stochastic model is available. We quantify and analyze how the induced uncertainty may be propagated through time as we collect new observations, and used to simultaneously provide robust estimates of the hidden signal and to learn the unknown parameters, via techniques based on pathwise filtering and new results on the optimal control of rough differential equations.

Certain polymer models are known to exhibit path localization in the sense that at low temperatures, the average fractional overlap of two independent samples from the Gibbs measure is bounded away from 0. Nevertheless, the question of where along the path this overlap takes place has remained unaddressed. In this article, we prove that on linear scales, overlap occurs along the entire length of the polymer. Namely, we consider time intervals of length $\mathit{\epsilon }N$, where $\mathit{\epsilon }>0$ is fixed but arbitrarily small. We then identify a constant number of distinguished trajectories such that the Gibbs measure is concentrated on paths having, with one of these distinguished paths, a fixed positive overlap simultaneously in every such interval. This result is obtained in all dimensions for a Gaussian random environment by using a recent non-local result as a key input.

We consider the asymmetric simple exclusion process (ASEP) on $\mathbb{Z}$. For continuous densities, ASEP is in local equilibrium for large times, at discontinuities however, one expects to see a dynamical phase transition, i.e. a mixture of different equilibriums. We consider ASEP with deterministic initial data such that at large times, two rarefactions come together at the origin, and the density jumps from 0 to 1. Shifting the measure on the KPZ $1∕3$ scale, we show that the law of ASEP converges to a mixture of the Dirac measures with only holes resp. only particles. The parameter of that mixture is the probability that the second class particle, which is distributed as the difference of two independent GUEs, stays to the left of the shift. This should be compared with the results of Ferrari and Fontes from 1994 [6], who obtained a mixture of Bernoulli product measures at discontinuities created by random initial data, with the GUEs replaced by Gaussians.

We generalise the integration by parts formulae obtained in a recent work with Zambotti to Bessel bridges on $[0,1]$ with arbitrary boundary values, as well as Bessel processes with arbitrary initial conditions. This allows us to write, formally, the corresponding dynamics using renormalised local times, thus extending the Bessel SPDEs of [7] to general Dirichlet boundary conditions. We also prove a dynamical result for the case of dimension 2, by providing a construction of the stationary dynamics corresponding to the law of a 2-dimensional Bessel bridge.

Three concepts of local times for deterministic càdlàg paths are developed and the corresponding pathwise Tanaka–Meyer formulae are provided. For semimartingales, it is shown that their sample paths a.s. satisfy all three pathwise definitions of local times and that all coincide with the classical semimartingale local time. In particular, this demonstrates that each definition constitutes a legit pathwise counterpart of probabilistic local times. The last pathwise construction presented in the paper expresses local times in terms of normalized numbers of interval crossings and does not depend on the choice of the sequence of grids. This is a new result also for càdlàg semimartingales, which may be related to previous results of Nicole El Karoui [11] and Marc Lemieux [23].

We extend the hypocoercivity framework for piecewise-deterministic Markov process (PDMP) Monte Carlo established in [2] to heavy-tailed target distributions, which exhibit subgeometric rates of convergence to equilibrium. We make use of weak Poincaré inequalities, as developed in the work of [15], the ideas of which we adapt to the PDMPs of interest. On the way we report largely potential-independent approaches to bounding explicitly solutions of the Poisson equation of the Langevin diffusion and its first and second derivatives, required here to control various terms arising in the application of the hypocoercivity result.

Given a representation of a compact Lie group and a state we define a probability measure on the coadjoint orbits of the dominant weights by considering the decomposition into irreducible components. For large tensor powers and independent copies of the state we show that the induced probability distributions converge to the value of the moment map. For faithful states we prove that the measures satisfy the large deviation principle with an explicitly given rate function.

We study two models of growing recursive trees. For both models, the tree initially contains a single vertex $u_1$ and at each time $n\ge 2$ a new vertex $u_n$ is added to the tree and its parent is chosen randomly according to some rule. In the weighted recursive tree, we choose the parent $u_k$ of $u_n$ among $\{u_1,u_2,\dots ,u_{n-1}\}$ with probability proportional to $w_k$, where ${(w_n)}_{n\ge 1}$ is some deterministic sequence that we fix beforehand. In the affine preferential attachment tree with fitnesses, the probability of choosing any $u_k$ is proportional to $a_k+{deg}^{+}(u_k)$, where ${deg}^{+}(u_k)$ denotes its current number of children, and the sequence of fitnesses ${(a_n)}_{n\ge 1}$ is deterministic and chosen as a parameter of the model.

We show that for any sequence ${(a_n)}_{n\ge 1}$, the corresponding preferential attachment tree has the same distribution as some weighted recursive tree with a random sequence of weights (with some explicit distribution). We then prove almost sure scaling limit convergences for some statistics associated with weighted recursive trees as time goes to infinity, such as degree sequence, height, profile and also the weak convergence of some measures carried on the tree. Thanks to the connection between the two models, these results also apply to affine preferential attachment trees.

We construct the ${\mathrm{\Phi }}_3^4$ measure on a periodic three dimensional box as an absolutely continuous perturbation of a random translation of the Gaussian free field. The shifted measure is constructed via Girsanov’s theorem and the relevant filtration is the one generated by a scale parameter. As a byproduct we give a self-contained proof that the ${\mathrm{\Phi }}_3^4$ measure is singular wrt. the Gaussian free field.

The vertex-reinforced jump process (VRJP) is a form of self-interacting random walk in which the walker is biased towards returning to previously visited vertices with the bias depending linearly on the local time at these vertices. We prove that, for any initial bias, the weights sampled from the magic formula on a two-dimensional graph decay at least at a power-law rate. Via arguments of Sabot and Zeng, the result implies that the VRJP is recurrent in two dimensions for any initial bias.

This paper aims to provide some tools coming from functional inequalities to deal with quasi-stationarity for absorbed Markov processes. First, it is shown how a Poincaré inequality related to a suitable Doob transform entails exponential convergence of conditioned distributions to a quasi-stationary distribution in total variation and in 1-Wasserstein distance. A special attention is paid to multi-dimensional diffusion processes, for which the aforementioned Poincaré inequality is implied by an easier-to-check Bakry-Émery condition depending on the right eigenvector for the sub-Markovian generator, which is not always known. Under additional assumptions on the potential, it is possible to bypass this lack of knowledge showing that exponential quasi-ergodicity is entailed by the classical Bakry-Émery condition.

This paper is devoted to two different two-time-scale stochastic approximation algorithms for superquantile, also known as conditional value-at-risk, estimation. We shall investigate the asymptotic behavior of a Robbins-Monro estimator and its convexified version. Our main contribution is to establish the almost sure convergence, the quadratic strong law and the law of iterated logarithm for our estimates via a martingale approach. A joint asymptotic normality is also provided. Our theoretical analysis is illustrated by numerical experiments on real datasets.

Suppose N is a compact Riemannian manifold, in this paper we will introduce the definition of N-valued BSDE and $L^2({\mathbb{T}}^m;N)$-valued BSDE for which the solutions are not necessarily staying in only one local coordinate. Moreover, the global existence of a solution to $L^2({\mathbb{T}}^m;N)$-valued BSDE will be proved without any convexity condition on N.

We study an inhomogeneous random connection model in the connectivity regime. The vertex set of the graph is a homogeneous Poisson point process ${\mathcal{P}}_s$ of intensity $s>0$ on the unit cube $S={\left(-\frac{1}{2},\frac{1}{2}\right]}^d$, $d\ge 2$ . Each vertex is endowed with an independent random weight distributed as W, where $P(W>w)=w^{-\mathit{\beta }}{1}_{[1,\mathrm{\infty })}(w)$, $\mathit{\beta }>0$. Given the vertex set and the weights an edge exists between $x,y\in {\mathcal{P}}_s$ with probability $\left(1-exp\left(-\frac{\mathrm{\eta }{W}_x{W}_y}{{\left(d(x,y)∕r\right)}^{\mathit{\alpha }}}\right)\right),$ independent of everything else, where $\mathrm{\eta },\mathit{\alpha }>0$, $d(\cdot ,\cdot )$ is the toroidal metric on S and $r>0$ is a scaling parameter. We derive conditions on $\mathit{\alpha },\mathit{\beta }$ such that under the scaling $r_s{(\mathrm{\xi })}^d=\frac{1}{c_{0}s}\left(logs+(k-1)loglogs+\mathrm{\xi }+log\left(\frac{\mathit{\alpha }\mathit{\beta }}{k!d}\right)\right),$ $\mathrm{\xi }\in \mathbb{R}$, the number of vertices of degree k converges in total variation distance to a Poisson random variable with mean $e^{-\mathrm{\xi }}$ as $s\to \mathrm{\infty }$, where $c_0$ is an explicitly specified constant that depends on $\mathit{\alpha },\mathit{\beta },d$ and η but not on k. In particular, for $k=0$ we obtain the regime in which the number of isolated nodes stabilizes, a precursor to establishing a threshold for connectivity. We also derive a sufficient condition for the graph to be connected with high probability for large s. The Poisson approximation result is derived using the Stein’s method.

We consider a class of particle systems that generalizes the eigenvalues of a class of matrix-valued processes, of which the empirical measures converge to deterministic measures as the dimension goes to infinity. In this paper, we obtain central limit theorems (CLTs) to characterize the fluctuations of the empirical measures around the limit measures by using stochastic calculus. As applications, CLTs for Dyson’s Brownian motion and the eigenvalues of Wishart process are recovered under slightly more general initial conditions, and a CLT for the eigenvalues of a symmetric matrix-valued Ornstein-Uhlenbeck process is obtained.

We consider fixed point equations for probability measures charging measured compact metric spaces that naturally yield continuum random trees. On the one hand, we study the existence/uniqueness of the fixed points and the convergence of the corresponding iterative schemes. On the other hand, we study the geometric properties of the random measured real trees that are fixed points, in particular their fractal properties. We obtain bounds on the Minkowski and Hausdorff dimension, that are proved tight in a number of applications, including the very classical continuum random tree, but also for the dual trees of random recursive triangulations of the disk introduced by Curien and Le Gall [Ann Probab, vol. 39, 2011]. The method happens to be especially efficient to treat cases for which the mass measure on the real tree induced by natural encodings only provides weak estimates on the Hausdorff dimensions.

We study a novel general class of multidimensional type-I backward stochastic Volterra integral equations. Toward this goal, we introduce an infinite family of standard backward SDEs and establish its well-posedness, and we show that it is equivalent to that of a type-I backward stochastic Volterra integral equation. We also establish a representation formula in terms of non-linear semi-linear partial differential equation of Hamilton–Jacobi–Bellman type. As an application, we consider the study of time-inconsistent stochastic control from a game-theoretic point of view. We show the equivalence of two current approaches to this problem from both a probabilistic and an analytic point of view.

We study unilateral valuation problems for American options within the framework of a general nonlinear market by extending results from Bielecki et al. [9, 12] who examined contracts of European style. A BSDE approach is used to establish more explicit pricing, hedging and exercising results when solutions to reflected BSDEs have additional desirable properties. We employ for this purpose results on solutions to BSDEs and reflected BSDEs driven by RCLL martingales obtained by Nie and Rutkowski [62, 63].

We consider reflected backward stochastic differential equations with two optional barriers of class (D) satisfying Mokobodzki’s separation condition, and coefficient which is only continuous and non-increasing. We assume that data are merely integrable and the terminal time is an arbitrary (possibly infinite) stopping time. We study the problem of the existence and uniqueness of solutions to the mentioned equations, and their connections with the value process in nonlinear Dynkin games.

We consider an interacting particle system on the lattice involving pushing and blocking interactions, called PushASEP, in the presence of a wall at the origin. We show that the invariant measure of this system is equal in distribution to a vector of point-to-line last passage percolation times in a random geometrically distributed environment. The largest co-ordinates in both of these vectors are equal in distribution to the all-time supremum of a non-colliding random walk.

This note presents a method based on Feynman-Kac semigroups for logarithmic Sobolev inequalities. It follows the recent work of Bonnefont and Joulin on intertwining relations for diffusion operators, formerly used for spectral gap inequalities, and related to perturbation techniques. In particular, it goes beyond the Bakry-Émery criterion and allows to investigate high-dimensional effects on the optimal logarithmic Sobolev constant. The method is illustrated on particular examples (namely Subbotin distributions and double-well potentials), for which explicit dimension-free bounds on the latter constant are provided. We eventually discuss a brief comparison with the Holley-Stroock approach.

Recent works have shown that random triangulations decorated by critical ($p=1∕2$) Bernoulli site percolation converge in the scaling limit to a $\sqrt{8∕3}$-Liouville quantum gravity (LQG) surface (equivalently, a Brownian surface) decorated by ${\mathrm{SLE}}_6$ in two different ways:

We prove that one in fact has joint convergence in both of these two senses simultaneously. We also improve the metric convergence result by showing that the map decorated by the full collection of percolation interfaces (rather than just a single interface) converges to $\sqrt{8∕3}$-LQG decorated by ${\mathrm{SLE}}_6$ in the metric space sense.

This is the first work to prove simultaneous convergence of any random planar map model in the metric and peanosphere senses. Moreover, this work is an important step in an ongoing program to prove that random triangulations embedded into $\mathbb{C}$ via the so-called Cardy embedding converge to $\sqrt{8∕3}$-LQG.

Motivated by the study of a parasite infection in a cell line, we introduce a general class of Markov processes for the modelling of population dynamics. The population process evolves as a diffusion with positive jumps whose rate is a function of the population size. It also undergoes catastrophic events which kill a fraction of the population, at a rate depending on the population state. We study the long time behaviour of this class of processes.

Using a divergent Bass-Burdzy flow we construct a self-repelling one-dimensional diffusion. Heuristically, it can be interpreted as a solution to an SDE with a singular drift involving a derivative of the local time. We show that this self-repelling diffusion inverts the second Ray-Knight identity on the line. The proof goes through an approximation by a self-repelling jump processes that has been previously shown by the authors to invert the Ray-Knight identity in discrete.

Given a hypergraph $\mathcal{H}$, the $\mathcal{H}$-bootstrap process starts with an initial set of infected vertices of $\mathcal{H}$ and, at each step, a healthy vertex v becomes infected if there exists a hyperedge of $\mathcal{H}$ in which v is the unique healthy vertex. We say that the set of initially infected vertices percolates if every vertex of $\mathcal{H}$ is eventually infected. We show that this process exhibits a sharp threshold when $\mathcal{H}$ is a hypergraph obtained by randomly sampling hyperedges from an approximately d-regular r-uniform hypergraph satisfying some mild degree and codegree conditions; this confirms a conjecture of Morris. As a corollary, we obtain a sharp threshold for a variant of the graph bootstrap process for strictly 2-balanced graphs which generalises a result of Korándi, Peled and Sudakov. Our approach involves an application of the differential equations method.

In this paper we consider random walks on Galton-Watson trees with random conductances. On these trees, the distance of the walker to the root satisfies a law of large numbers with limit the effective velocity, or speed of the walk. We study the regularity of the speed as a function of the distribution of conductances, in particular when the distribution of conductances converges to a non-elliptic limit.

Consider an ergodic unimodular random one-ended planar graph $\mathtt{G}$ of finite expected degree. We prove that it has an isometry-invariant locally finite embedding in the Euclidean plane if and only if it is invariantly amenable. By “locally finite” we mean that any bounded open set intersects finitely many embedded edges. In particular, there exist invariant embeddings in the Euclidean plane for the Uniform Infinite Planar Triangulation and for the critical Augmented Galton-Watson Tree conditioned to survive. Roughly speaking, a unimodular embedding of $\mathtt{G}$ is one that is jointly unimodular with $\mathtt{G}$ when viewed as a decoration. We show that $\mathtt{G}$ has a unimodular embedding in the hyperbolic plane if it is invariantly nonamenable, and it has a unimodular embedding in the Euclidean plane if and only if it is invariantly amenable. Similar claims hold for representations by tilings instead of embeddings.

We consider a square random matrix of size N of the form $P(Y,A)$ where P is a noncommutative polynomial, A is a tuple of deterministic matrices converging in ∗-distribution, when N goes to infinity, towards a tuple a in some ${\mathcal{C}}^{\ast }$-probability space and Y is a tuple of independent matrices with i.i.d. centered entries with variance $1∕N$. We investigate the eigenvalues of $P(Y,A)$ outside the spectrum of $P(c,a)$ where c is a circular system which is free from a. We provide a sufficient condition to guarantee that these eigenvalues coincide asymptotically with those of $P(0,A)$.

Consider jump processes on a connected compact smooth Riemannian manifold, which are constructed by the canonical projection of the processes on the bundle of orthonormal frames. The condition under which the M-valued process is Markovian will be revisited as seen in Applebaum-Estrade [1]. Moreover, the gradient formula, which will be also called the integration by parts formula, can be also studied. The obtained formula can be regarded as the extended version of the celebrated Bismut formula on the case of diffusion processes.

We consider a 2D stochastic wave equation driven by a Gaussian noise, which is temporally white and spatially colored described by the Riesz kernel. Our first main result is the functional central limit theorem for the spatial average of the solution. And we also establish a quantitative central limit theorem for the marginal and the rate of convergence is described by the total-variation distance. A fundamental ingredient in our proofs is the pointwise $L^p$-estimate of Malliavin derivative, which is of independent interest.

Motivated by the goal of understanding the evolution of populations undergoing selection, we consider branching Brownian motion in which particles independently move according to one-dimensional Brownian motion with drift, each particle may either split into two or die, and the difference between the birth and death rates is a linear function of the position of the particle. We show that, under certain assumptions, after a sufficiently long time, the empirical distribution of the positions of the particles is approximately Gaussian. This provides mathematically rigorous justification for results in the biology literature indicating that the distribution of the fitness levels of individuals in a population over time evolves like a Gaussian traveling wave.

We study linearly edge-reinforced random walks on ${\mathbb{Z}}_{+}$, where each edge $\{x,x+1\}$ has the initial weight $x^{\mathit{\alpha }}$ for $x>0$ and 1 for $x=0$, and each time an edge is traversed, its weight is increased by Δ. It is known that the walk is recurrent if and only if $\mathit{\alpha }\le 1$. The aim of this paper is to study the almost sure behavior of the walk in the recurrent regime. For $\mathit{\alpha }<1$ and $\mathrm{\Delta }>0$, we obtain a limit theorem which is a counterpart of the law of the iterated logarithm for simple random walks. This reveals that the speed of the walk with $\mathrm{\Delta }>0$ is much slower than $\mathrm{\Delta }=0$. In the critical case $\mathit{\alpha }=1$, our (almost sure) bounds for the trajectory of the walk shows that there is a phase transition of the speed at $\mathrm{\Delta }=2$.

We prove the sharp bound for the expected spread upon opinions of $n\ge 2$ experts, who have access to different information sources represented by different σ-fields. Using symmetrization argument and direct combinatorial optimization we derive an explicit optimizer. Our results may turn out to be helpful not only for probabilists, but also for statisticians and economists.

Consider a system of n weakly interacting particles driven by independent Brownian motions. In many instances, it is well known that the empirical measure converges to the solution of a partial differential equation, usually called McKean-Vlasov or Fokker-Planck equation, as n tends to infinity. We propose a relatively new approach to show this convergence by directly studying the stochastic partial differential equation that the empirical measure satisfies for each fixed n. Under a suitable control on the noise term, which appears due to the finiteness of the system, we are able to prove that the stochastic perturbation goes to zero, showing that the limiting measure is a solution to the classical McKean-Vlasov equation. In contrast with known results, we do not require any independence or finite moment assumption on the initial condition, but the only weak convergence.

The evolution of the empirical measure is studied in a suitable class of Hilbert spaces where the noise term is controlled using two distinct but complementary techniques: rough paths theory and maximal inequalities for self-normalized processes.

Let $H:=-\frac{1}{2}\mathrm{\Delta }+V$ be a one-dimensional continuum Schrödinger operator. Consider $\hat{H}:=H+\mathrm{\xi }$, where ξ is a translation invariant Gaussian noise. Under some assumptions on ξ, we prove that if V is locally integrable, bounded below, and grows faster than log at infinity, then the semigroup ${\mathrm{e}}^{-t\hat{H}}$ is trace class and admits a probabilistic representation via a Feynman-Kac formula. Our result applies to operators acting on the whole line $\mathbb{R}$, the half line $(0,\mathrm{\infty })$, or a bounded interval $(0,b)$, with a variety of boundary conditions. Our method of proof consists of a comprehensive generalization of techniques recently developed in the random matrix theory literature to tackle this problem in the special case where $\hat{H}$ is the stochastic Airy operator.

We study a class of interacting particle systems with asymmetric interaction showing a self-duality property. The class includes the ASEP($q,\mathit{\theta }$), asymmetric exclusion process, with a repulsive interaction, allowing up to $\mathit{\theta }\in \mathbb{N}$ particles in each site, and the ASIP$(q,\mathit{\theta })$, $\mathit{\theta }\in {\mathbb{R}}^{+}$, asymmetric inclusion process, that is its attractive counterpart. We extend to the asymmetric setting the investigation of orthogonal duality properties done in [8] for symmetric processes. The analysis leads to multivariate $q-$analogues of Krawtchouk polynomials and Meixner polynomials as orthogonal duality functions for the generalized asymmetric exclusion process and its asymmetric inclusion version, respectively. We also show how the q-Krawtchouk orthogonality relations can be used to compute exponential moments and correlations of ASEP($q,\mathit{\theta }$).

The Horton–Strahler number of a tree is a measure of its branching complexity. It is also known in the literature as the register function. We show that for critical Galton–Watson trees with finite variance, conditioned to be of size n, the Horton–Strahler number grows as $\frac{1}{2}{log}_{2}n$ in probability. We further define some generalizations of this number. Among these are the rigid Horton–Strahler number and the k-ary register function, for which we prove asymptotic results analogous to the standard case.

The range process $R_n$ of a random walk is the collection of sites visited by the random walk up to time n. In this paper we deal with the question of whether the range process of a random walk or the range process of a cocycle over an ergodic transformation is almost surely a Følner sequence and show the following results: (a) The size of the inner boundary $|\partial R_n|$ of the range of recurrent aperiodic random walks on ${\mathbb{Z}}^2$ with finite variance and aperiodic random walks in $\mathbb{Z}$ in the standard domain of attraction of the Cauchy distribution, divided by $n∕{log}^2(n)$, converges to a constant almost surely. (b) We establish a formula for the Følner asymptotic of transient cocycles over an ergodic probability preserving transformation and use it to show that for admissible transient random walks on finitely generated groups, the range is never a Følner sequence unless the walk is a skip-free random walk on $\mathbb{Z}$. (c) For strongly aperiodic random walks in the domain of attraction of symmetric α-stable distributions with $1<\mathit{\alpha }\le 2$, we prove a sharp polynomial upper bound for the decay at infinity of $|\partial R_n|∕|R_n|$. This last result shows that the range process of these random walks is almost surely a Følner sequence.

We introduce an explicit Milstein-type scheme for McKean–Vlasov stochastic differential equations using the notion of a measure derivative given by P.-L. Lions in his lectures at the Collège de France and presented in [9]. We show that the proposed Milstein-type scheme converges to the true solution in strong sense with a rate equal to 1.0. The drift coefficient is allowed to grow super-linearly in the state variable and both the drift and the diffusion coefficients are assumed to be only once differentiable in variables corresponding to state and measure. Furthermore, derivatives of drift and diffusion coefficients with respect to the measure component are uniformly bounded. The challenges arising due to the dependence of coefficients on measures are tackled and our findings are consistent with the analogous results for stochastic differential equations.

We consider the spread of a supercritical stochastic $\mathit{SIR}$ (Susceptible, Infectious, Recovered) epidemic on a configuration model random graph. We mainly focus on the final stages of a large outbreak and provide limit results for the duration of the entire epidemic, while we allow for non-exponential distributions of the infectious period and for both finite and infinite variance of the asymptotic degree distribution in the graph.

Our analysis relies on the analysis of some subcritical continuous time branching processes and on ideas from first passage percolation.

As an application we investigate the effect of vaccination with an all-or-nothing vaccine on the duration of the epidemic. We show that if vaccination fails to prevent the epidemic, it often – but not always – increases the duration of the epidemic.

We provide two methods to construct zero-range processes with superlinear rates on ${\mathbb{Z}}^d$. In the first method these rates can grow very fast, if either the dynamics and the initial distribution are translation invariant or if only nearest neighbour translation invariant jumps are permitted, in the one-dimensional lattice. In the second method the rates cannot grow as fast but more general dynamics are allowed.

We investigate the probabilistic and analytic properties of Volterra processes constructed as pathwise integrals of deterministic kernels with respect to the Hölder continuous trajectories of Hilbert-valued Gaussian processes. To this end, we extend the Volterra sewing lemma from [18] to the two dimensional case, in order to construct two dimensional operator-valued Volterra integrals of Young type. We prove that the covariance operator associated to infinite dimensional Volterra processes can be represented by such a two dimensional integral, which extends the current notion of representation for such covariance operators. We then discuss a series of applications of these results, including the construction of a rough path associated to a Volterra process driven by Gaussian noise with possibly irregular covariance structures, as well as a description of the irregular covariance structure arising from Gaussian processes time-shifted along irregular trajectories. Furthermore, we consider an infinite dimensional fractional Ornstein-Uhlenbeck process driven by Gaussian noise, which can be seen as an extension of the volatility model proposed by Rosenbaum et al. in [13].

In [24] a functional limit theorem was proved. It states that symmetric processes associated with resistance metric measure spaces converge when the underlying spaces converge with respect to the Gromov-Hausdorff-vague topology, and a certain uniform recurrence condition is satisfied. Such a theorem finds particularly nice applications if the resistance metric measure space is a metric measure tree. To illustrate this, we state functional limit theorems in old and new examples of suitably rescaled random walks in random environment on trees.

First, we take a critical Galton-Watson tree conditioned on its total progeny and a non-lattice branching random walk on ${\mathbb{R}}^d$ indexed by it. Then, conditional on that, we consider a biased random walk on the range of the preceding. Here, by non-lattice we mean that distinct branches of the tree do not intersect once mapped in ${\mathbb{R}}^d$. This excludes the possibility that the random walk on the range may jump from one branch to the other without returning to the most recent common ancestor. We prove, after introducing the bias parameter ${\mathit{\beta }}^{n^{-1∕4}}$, for some $\mathit{\beta }>1$, that the biased random walk on the range of a large critical non-lattice branching random walk converges to a Brownian motion in a random Gaussian potential on Aldous’ continuum random tree (CRT).

Our second new result introduces the scaling limit of the edge-reinforced random walk on a size-conditioned Galton-Watson tree with finite variance as a Brownian motion in a random Gaussian potential on the CRT with a drift proportional to the distance to the root.

In this work we derive limit theorems for trawl processes. First, we study the asymptotic behavior of the partial sums of the discretized trawl process ${(X_{i{\mathrm{\Delta }}_n})}_{i=0}^{\lfloor nt\rfloor -1}$, under the assumption that as $n\uparrow \mathrm{\infty }$, ${\mathrm{\Delta }}_n\downarrow 0$ and $n{\mathrm{\Delta }}_n\to \mathrm{\mu }\in [0,+\mathrm{\infty }]$. Second, we prove a general result on functional convergence in distribution of trawl processes. As an application of this result, we show that a trawl process whose Lévy measure tends to infinity converges in distribution, under suitable rescaling, to a Gaussian moving average process.

In this paper we study the self-similar processes with stationary increments in a discrete-time setting. Different from the continuous-time case, it is shown that the scaling function of such a process may not take the form of a power function $b(a)=a^H$. More precisely, its scaling function can belong to one of three types, among which one type is degenerate, one type has a continuous-time counterpart, while the other type is new and unique for the discrete-time setting. We then focus on this last type of processes, construct two classes of examples, and prove a special spectral representation result for the processes of this type. We also derive basic properties of discrete-time self-similar processes with stationary increments of different types.

In the q-voter model, the voter at x changes its opinion at rate $f_x^q$, where $f_x$ is the fraction of neighbors with the opposite opinion. Mean-field calculations suggest that there should be coexistence between opinions if $q<1$ and clustering if $q>1$. This model has been extensively studied by physicists, but we do not know of any rigorous results. In this paper, we use the machinery of voter model perturbations to show that the conjectured behavior holds for q close to 1. More precisely, we show that if $q<1$, then for any $m<\mathrm{\infty }$ the process on the three-dimensional torus with n points survives for time $n^m$, and after an initial transient phase has a density that it is always close to 1/2. Readers familiar with long time survival results for the contact process and other praticle systems might expect the conjecture to say survival occurs for time $exp(\mathit{\gamma }n)$ with $\mathit{\gamma }>0$, however we show persistence does not hold for $exp(n^{\mathit{\beta }})$ with $\mathit{\beta }>1∕3$. If $q>1$, then the process rapidly reaches fixation on one opinion. It is interesting to note that in the second case the limiting ODE (on its sped up time scale) reaches 0 at time $logn$ but the stochastic process on the same time scale dies out at time $(1∕3)logn$.

This paper provides an extended case study of the cutoff phenomenon for a prototypical class of nonlinear Langevin systems with a single stable state perturbed by an additive pure jump Lévy noise of small amplitude $\mathit{\epsilon }>0$, where the driving noise process is of layered stable type. Under a drift coercivity condition the associated family of processes $X^{\mathit{\epsilon }}$ turns out to be exponentially ergodic with equilibrium distribution ${\mathrm{\mu }}^{\mathit{\epsilon }}$ in total variation distance which extends a result from [60] to arbitrary polynomial moments.

The main results establish the cutoff phenomenon with respect to the total variation, under a sufficient smoothing condition of Blumenthal-Getoor index $\mathit{\alpha }>\frac{3}{2}$. That is to say, in this setting we identify a deterministic time scale ${\mathfrak{t}}_{\mathit{\epsilon }}^{\mathrm{cut}}$ satisfying ${\mathfrak{t}}_{\mathit{\epsilon }}^{\mathrm{cut}}\to \mathrm{\infty }$, as $\mathit{\epsilon }\to 0$, and a respective time window, ${\mathfrak{t}}_{\mathit{\epsilon }}^{\mathrm{cut}}\pm o({\mathfrak{t}}_{\mathit{\epsilon }}^{\mathrm{cut}})$, during which the total variation distance between the current state and its equilibrium ${\mathrm{\mu }}^{\mathit{\epsilon }}$ essentially collapses as ε tends to zero. In addition, we extend the dynamical characterization under which the latter phenomenon can be described by the convergence of such distance to a unique profile function first established in [9] to the Lévy case for nonlinear drift. This leads to sufficient conditions, which can be verified in examples, such as gradient systems subject to small symmetric α-stable noise for $\mathit{\alpha }>\frac{3}{2}$. The proof techniques differ completely from the Gaussian case due to the absence of a respective Girsanov transform which couples the nonlinear equation and the linear approximation asymptotically even for short times.

Let f be a martingale with values in a uniformly p-smooth Banach space and w any positive weight. We show that $\mathbb{E}(f^{\ast }\cdot w)\lesssim \mathbb{E}(S_{p}f\cdot w^{\ast })$, where ${\cdot }^{\ast }$ is the martingale maximal operator and $S_p$ is the ${\ell }^p$ sum of martingale increments.

We study a family of non-linear McKean-Vlasov SDEs driven by a Poisson measure, modelling the mean-field asymptotic of a network of generalized Integrate-and-Fire neurons. We give sufficient conditions to have periodic solutions through a Hopf bifurcation. Our spectral conditions involve the location of the roots of an explicit holomorphic function. The proof relies on two main ingredients. First, we introduce a discrete time Markov Chain modeling the phases of the successive spikes of a neuron. The invariant measure of this Markov Chain is related to the shape of the periodic solutions. Secondly, we use the Lyapunov-Schmidt method to obtain self-consistent oscillations. We illustrate the result with a toy model for which all the spectral conditions can be analytically checked.

Let ${(X_i)}_{i\ge 1}$ be i.i.d. random variables with $\mathsf{E}{X}_1=0$, regularly varying with exponent $a>2$ and $t^{a}P(|X_1|>t)\sim L(t)$ slowly varying as $t\to \mathrm{\infty }$. We give the limit distribution of $T_n(\mathit{\gamma })={max}_{0\le j<k\le n}|X_{j+1}+\cdots +X_k|{(k-j)}^{-\mathit{\gamma }}$ in the threshold case ${\mathit{\gamma }}_a:=1∕2-1∕a$ which separates the Brownian phase corresponding to $0\le \mathit{\gamma }<{\mathit{\gamma }}_a$ where the limit of $T_n(\mathit{\gamma })$ is $\mathit{\sigma }T(\mathit{\gamma })$, with ${\mathit{\sigma }}^2=\mathsf{E}{X}_1^2$, $T(\mathit{\gamma })$ is the γ-Hölder norm of a standard Brownian motion and the Fréchet phase corresponding to ${\mathit{\gamma }}_a<\mathit{\gamma }<1$ where the limit of $T_n(\mathit{\gamma })$ is $Y_a$ with Fréchet distribution $P(Y_a\le x)=exp(-x^{-a})$, $x>0$. We prove that $c_n^{-1}(T_n({\mathit{\gamma }}_a)-{\mathrm{\mu }}_n)$, converges in distribution to some random variable Z if and only if L has a limit ${\mathit{\tau }}^a\in [0,\mathrm{\infty }]$ at infinity. In such case, there are $A>0$, $B\in \mathbb{R}$ such that $Z=A{V}_{a,\mathit{\sigma },\mathit{\tau }}+B$ in distribution, where for $0<\mathit{\tau }<\mathrm{\infty }$, $V_{a,\mathit{\sigma },\mathit{\tau }}:=max(\mathit{\sigma }T({\mathit{\gamma }}_a),\mathit{\tau }{Y}_a)$ with $T({\mathit{\gamma }}_a)$ and $Y_a$ independent and $V_{a,\mathit{\sigma },0}:=\mathit{\sigma }T({\mathit{\gamma }}_a)$, $V_{a,\mathit{\sigma },\mathrm{\infty }}:=Y_a$. When $\mathit{\tau }<\mathrm{\infty }$, a possible choice for the normalization is $c_n=n^{-1∕a}$ and ${\mathrm{\mu }}_n=0$, with $Z=V_{a,\mathit{\sigma },\mathit{\tau }}$. We also build an example where L has no limit at infinity and ${(T_n(\mathit{\gamma }))}_{n\ge 1}$ has for each $\mathit{\tau }\in [0,\mathrm{\infty }]$ a subsequence converging after normalization to $V_{a,\mathit{\sigma },\mathit{\tau }}$.