The paper is devoted to establishing some general exponential inequalities for supermartingales. The inequalities improve or generalize many exponential inequalities of Bennett, Freedman, de la Peña, Pinelis and van de Geer. Moreover, our concentration inequalities also improve some known inequalities for sums of independent random variables. Applications associated with linear regressions, autoregressive processes and branching processes are also provided.

The loop clusters of a Poissonian ensemble of Markov loops on a finite or countable graph have been studied in \cite{Markovian-loop-clusters-on-graphs}. In the present article, we study the loop clusters associated with a rotation invariant nearest neighbor walk on the discrete circle $G^{(n)}$ with $n$ vertices. We prove a convergence result of the loop clusters on $G^{(n)}$, as $n\rightarrow\infty$, under suitable condition of the parameters. These parameters are chosen in such a way that the rotation invariant nearest neighbor walk on $G^{(n)}$, as $n\rightarrow\infty$, converges to a Brownian motion on circle $\mathbb{S}^{1}=\mathbb{R}/\mathbb{Z}$ with certain drift and killing rate. In the final section, we show that several limit results are predicted by Brownian loop-soup on $\mathbb{S}^{1}$.

We consider dynamic random walks where the nearest neighbour jump rates are determined by an underlying supercritical contact process in equilibrium. This has previously been studied by den Hollander and dos Santos (arXiv:1209.1511). We show the CLT for such a random walk, valid for all supercritical infection rates for the contact process environment.

We prove general limit theorems for sums of functions of subtrees of (random) binary search trees and random recursive trees. The proofs use a new version of a representation by Devroye, and Stein's method for both normal and Poisson approximation together with certain couplings.

As a consequence, we give simple new proofs of the fact that the number of fringe trees of size $ k=k_n $ in the binary search tree or in the random recursive tree (of total size $ n $) has an asymptotical Poisson distribution if $ k\rightarrow\infty $, and that the distribution is asymptotically normal for $ k=o(\sqrt{n}) $. Furthermore, we prove similar results for the number of subtrees of size $ k $ with some required property $ P $, e.g., the number of copies of a certain fixed subtree $ T $. Using the Cramér-Wold device, we show also that these random numbers for different fixed subtrees converge jointly to a multivariate normal distribution. <br /><br />We complete the paper by giving examples of applications of the general results, e.g., we obtain a normal limit law for the number of $ \ell $-protected nodes in a binary search tree or in a random recursive tree.

We consider refined versions of Markov chains related to juggling introduced by Warrington. We further generalize the construction to juggling with arbitrary heights as well as infinitely many balls, which are expressed more succinctly in terms of Markov chains on integer partitions. In all cases, we give explicit product formulas for the stationary probabilities and closed-form expressions for the normalization factor. We also refine and generalize enriched Markov chains on set partitions. Lastly, we prove that in one case, the stationary distribution is attained in finite time.

We study the depinning transition of the $1+1$ dimensional directed polymer in a random environment with a defect line. The random environment consists of i.i.d. potential values assigned to each site of $\mathbb{Z}^2$; sites on the positive axis have the potential enhanced by a deterministic value $u$. We show that for small inverse temperature $\beta$ the quenched and annealed free energies differ significantly at most in a small neighborhood (of size of order $\beta$) of the annealed critical point $u_c^a=0$. For the case $u=0$, we show that the difference between quenched and annealed free energies is of order $\beta^4$ as $\beta\to 0$, assuming only finiteness of exponential moments of the potential values, improving existing results which required stronger assumptions.

It has been established under very general conditions that the ergodic properties of Markov processes are inherited by their conditional distributions given partial information. While the existing theory provides a rather complete picture of classical filtering models, many infinite dimensional problems are outside its scope. Far from being a technical issue, the infinite dimensional setting gives rise to surprising phenomena and new questions in filtering theory. The aim of this paper is to discuss some elementary examples, conjectures, and general theory that arise in this setting, and to highlight connections with problems in statistical mechanics and ergodic theory. In particular, we exhibit a simple example of a uniformly ergodic model in which ergodicity of the filter undergoes a phase transition, and we develop some qualitative understanding as to when such phenomena can and cannot occur.We also discuss closely related problems in the setting of conditional Markov random fields.

We consider a generalization of the classical pinning problem for integer-valued random walks conditioned to stay non-negative. More specifically, we take pinning potentials of the form $\sum_{j\geq 0}\epsilon_j N_j$, where $N_j$ is the number of visits to the state $j$ and $\{\epsilon_j\}$ is a non-negative sequence. Partly motivated by similar problems for low-temperature contour models in statistical physics, we aim at finding a sharp characterization of the threshold of the wetting transition, especially in the regime where the variance $\sigma^2$ of the single step of the random walk is small. Our main result says that, for natural choices of the pinning sequence $\{\epsilon_j\}$, localization (respectively delocalization) occurs if $\sigma^{-2}\sum_{ j\geq0}(j+1)\epsilon_j\geq\delta^{-1}$ (respectively $\le \delta$), for some universal $\delta < 1$. Our finding is reminiscent of the classical Bargmann-Jost-Pais criteria for the absence of bound states for the radial Schrödinger equation. The core of the proof is a recursive argument to bound the free energy of the model. Our approach is rather robust, which allows us to obtain similar results in the case where the random walk trajectory is replaced by a self-avoiding path $\gamma$ in $\mathbb Z^2$ with weight $\exp(-\beta |\gamma|)$, $|\gamma|$ being the length of the path and $\beta > 0$ a large enough parameter. This generalization is directly relevant for applications to the above mentioned contour models.

We study protected nodes in $m-$ary search trees, by putting them in context of generalized Pólya urns. We show that the number of two-protected nodes (the nodes that are neither leaves nor parents of leaves) in a random ternary search tree is asymptotically normal. The methods apply in principle to $m-$ary search trees with larger $m$ as well, although the size of the matrices used in the calculations grow rapidly with $m$; we conjecture that the method yields an asymptotically normal distribution for all $m \leq 26$.

The one-protected nodes, and their complement, i.e., the leaves, are easier to analyze. By using a simpler urn (that is similar to the one that has earlier been used to study the total number of nodes in $m-$ary search trees), we prove normal limit laws for the number of one-protected nodes and the number of leaves for all $m \leq 26$.

We give estimates for the probability that a chordal, radial or two-sided radial SLE$_\kappa$ curve retreats far from its terminal point after coming close to it, for $\kappa \le 4$. The estimates are uniform over all initial segments of the curve, and are sharp up to a universal constant.

A random $n$-permutation may be generated by sequentially removing random cards $C_1,...,C_n$ from an $n$-card deck $D = \{1,...,n\}$. The permutation $\sigma$ is simply the sequence of cards in the order they are removed. This permutation is itself uniformly random, as long as each random card $C_t$ is drawn uniformly from the remaining set at time $t$. We consider, here, a variant of this simple procedure in which one is given a choice between $k$ random cards from the remaining set at each step, and selects the lowest numbered of these for removal. This induces a bias towards selecting lower numbered of the remaining cards at each step, and therefore leads to a final permutation which is more ''ordered'' than in the uniform case (i.e. closer to the identity permutation id $=(1,2,3,...,n)$).

We quantify this effect in terms of two natural measures of order: The number of inversions $I$ and the length of the longest increasing subsequence $L$. For inversions, we establish a weak law of large numbers and central limit theorem, both for fixed and growing $k$. For the longest increasing subsequence, we establish the rate of scaling, in general, and existence of a weak law in the case of growing $k$. We also show that the minimum strategy, of selecting the minimum of the $k$ given choices at each step, is optimal for minimizing the number of inversions in the space of all online $k$-card selection rules.

We deal with a stochastic control problem subject to a stochastic variational inequality with delay. By deriving the adjoint equation as an anticipated backward stochastic differential equation, we are able to establish necessary conditions of optimality under the form of a Pontryagin-Bensoussan stochastic maximum principle. This is achieved first for cadlag controls, by explicitly writing the coefficients of the adjoint equation in terms of the local time of the state process. The general result is then obtained by approximating the optimal control with continuous controls and applying Ekeland's variational principle to the approximating sequence.

Consider two unit balls in a $d$-dimensional flat torus with edge length $r$, for $d\geq 2$. The balls do not move by themselves but they are pushed by a Brownian motion. The balls never intersect---they reflect if they touch. It is proved that the joint distribution of the processes representing the centers of the balls converges to the distribution of two independent Brownian motions when $r\to \infty$, assuming that we use a proper clock and proper scaling. The diffusion coefficient of the limit process depends on the dimension. The positions of the balls are asymptotically independent also in the following sense. The rescaled stationary distributions of the centers of the balls converge to the product of the stationary (hence uniform) distributions for each ball separately, as $r\to\infty$.

Let $(\xi_i)_{i \in {\mathbb Z}}$ be a stationary Harris recurrent geometrically erodic Markov chain on a countably generated state space $(E, {\mathcal B})$. Let $f$ be a bounded and measurable function from $E$ into ${\mathbb R}$ satisfying the condition ${\mathbb E} ( f ( \xi_0))=0$. In this paper, we obtain the almost sure strong approximation of the partial sums $S_n(f) = \sum_{i=1}^n f ( \xi_i)$ by the partial sums of a sequence of independent and identically distributed Gaussian random variables with the optimal rate $O ( \log n)$.

We consider multiple time scales systems of stochastic differential equations with small noise in random environments. We prove a quenched large deviations principle with explicit characterization of the action functional. The random medium is assumed to be stationary and ergodic. In the course of the proof we also prove related quenched ergodic theorems for controlled diffusion processes in random environments that are of independent interest. The proof relies entirely on probabilistic arguments, allowing to obtain detailed information on how the rare event occurs. We derive a control, equivalently a change of measure, that leads to the large deviations lower bound. This information on the change of measure can motivate the design of asymptotically efficient Monte Carlo importance sampling schemes for multiscale systems in random environments

We give a new, simple construction of the $\alpha$-stable tree for $\alpha \in (1,2]$. We obtain it as the closure of an increasing sequence of $\mathbb{R}$-trees inductively built by gluing together line-segments one by one. The lengths of these line-segments are related to the the increments of an increasing $\mathbb{R}_+$-valued Markov chain. For $\alpha = 2$, we recover Aldous' line-breaking construction of the Brownian continuum random tree based on an inhomogeneous Poisson process.

In this paper we investigate the dynamical behavior of an interface or polymer, in interaction with a distant attractive substrate. The interface is modeled by the graph of a nearest neighbor path with non-negative integer coordinates, and the equilibrium measure associates to each path $\eta$ a probability proportional to $\lambda^{H(\eta)}$ where $\lambda$ is non-negative and $H(\eta)$ is the number of contacts between $\eta$ and the substrate at zero. The dynamics is the natural "spin flip" dynamics associated to this equilibrium measure. We let the distance to the substrate at both polymer ends be equal to $aN$, where $a \in (0,1/2)$ is a fixed parameter, and $N$ is the length the system. With this setup, we show that the dynamical behavior of the system crucially depends on $\lambda$: when $\lambda \leq 2/(1-2a)$ we show that the system only needs a time which is polynomial in $N$ to reach its equilibrium state, whereas if $\lambda > 2/(1-2a)$ the mixing time is exponential in $N$ and the system relaxes in an exponential manner which is typical of metastability.

We consider the branching random walk $(Z_n)_{n \geq 0}$ on $\mathbb{R}$ where the underlying motion is of a simple random walk and branching is at least binary and at most decaying exponentially in law. For a large class of measurable sets $A \subseteq \mathbb{R}$, it is well known that $\overline{Z}_n(\sqrt{n} A) \to \nu(A)$ almost surely as $n \to \infty$, where $\overline{Z}_n$ is the particles empirical distribution at generation $n$ and $\nu$ is the standard Gaussian measure on $\mathbb{R}$. We therefore analyze the rate at which $\mathbb{P}(\overline{Z}_n(\sqrt{n}A) > \nu(A) + \epsilon)$ and $\mathbb{P}(\overline{Z}_n(\sqrt{n}A) < \nu(A) - \epsilon)$ go to zero for any $\epsilon > 0$. We show that the decay is doubly exponential in either $n$ or $\sqrt{n}$, depending on $A$ and $\epsilon$ and find the leading coefficient in the top exponent. To the best of our knowledge, this is the first time such large deviation probabilities are treated in this model.

We study parabolic stochastic partial differential equations(SPDEs), driven by two types of operators: one linear closedoperator generating a $C_0-$semigroup and one linear boundedoperator with Wick-type multiplication, all of them set in theinfinite dimensional space framework of white noise analysis. Weprove existence and uniqueness of solutions for this class of SPDEs.In particular, we also treat the stationary case when thetime-derivative is equal to zero.

We obtain necessary and sufficient conditions for the regular variation of the variance of partial sums of functionals of discrete and continuous-time stationary Markov processes with normal transition operators. We also construct a class of Metropolis-Hastings algorithms which satisfy a central limit theorem and invariance principle when the variance is not linear in $n$

By using Bismut's approach about the Malliavin calculus with jumps, we study the regularity of the distributional density for SDEs driven by degenerate additive Lévy noises. Under full Hörmander's conditions, we prove the existence of distributional density and the weak continuity in the first variable of the distributional density.Under the uniform first order Lie's bracket condition, we also prove the smoothness of the density.

We consider the stochastic recursion $X_{n+1} = M_{n+1}X_n + Q_{n+1}$ on $\mathbb{R}^d$,

where ($M_n, Q_n$) are i.i.d. random variables such that $Q_n$ are translations, $M_n$ are similarities of the Euclidean space $\mathbb{R}^d$. Under some standard assumptions the sequence $X_n$ converges to a random variable $R$ and the law $\nu$ of $R$ is the unique stationary measure of the process. Moreover,

the weak limit of properly dilated measure $\nu$ exists, defining thus a homogeneous tail measure $\Lambda$. In this paper we study the rate of convergence of dilations of $\nu$ to $\Lambda$

In particular in the one dimensional setting, when $(M_n,Q_n) \in \mathbb{R}^+\times \mathbb{R}$, $\mathbb{E} M_n^{\alpha }=1$ and $X_n\in \mathbb{R}$, the Kesten renewal theorem says that $t^\alpha\mathbb{P}[|R|>t]$ converges to some strictly positive constant $C_+$. Our main result says that $$\big|t^\alpha\mathbb{P}[|R|>t]-C_+\big|\le C (\log t)^{-\sigma},$$ for some $\sigma>0$ and large $t$. It generalizes the previous one by Goldie.

The spherical ensemble is a well-studied determinantal process with a fixed number of points on $\mathbb{S}^2$. The points of this process correspond to the generalized eigenvalues of two appropriately chosen random matrices, mapped to the surface of the sphere by stereographic projection. This model can be considered as a spherical analogue for other random matrix models on the unit circle and complex plane such as the circular unitary ensemble or the Ginibre ensemble, and is one of the most natural constructions of a (statistically) rotation invariant point process with repelling property on the sphere. In this paper we study the spherical ensemble and its local repelling property by investigating the minimum spacing between the points and the area of the largest empty cap. Moreover, we consider this process as a way of distributing points uniformly on the sphere. To this aim, we study two "metrics" to measure the uniformity of anarrangement of points on the sphere. For each of these metrics (discrepancy and Riesz energies) we obtain some bounds and investigate the asymptotic behavior when the number of points tends to infinity. It is remarkable that though the model is random, because of the repelling property of the points, the behavior can be proved to be as good as the best known constructions (for discrepancy) or even better than the best known constructions (for Riesz energies).

It is conjectured that in dimensions d ≥ 2 any random walk in an i.i.d. uniformly elliptic random environment (RWRE) which is directionally transient is ballistic. The ballisticity conditions for RWRE somehow interpolate between directional transience and ballisticity and have served to quantify the gap which would need to be proven in order to answer affirmatively this conjecture. Two important ballisticity conditions introduced by Sznitman in 2001 and 2002 are the so called conditions (T′) and (T): given a slab of width L orthogonal to l, condition (T′) in direction l is the requirement that the annealed exit probability of the walk through the side of the slab in the half-space ${x : x \cdot l < 0}$, decays faster than $e(−CL^\gamma)$ for all $\gamma\in (0, 1)$ and some constant C &gt; 0, while condition (T) in direction l is the requirement that the decay is exponential e(−CL). It is believed that (T′) implies (T). In this article we show that (T′) implies at least an almost (in a sense to be made precise) exponential decay.

We establish Tracy-Widom asymptotics for the partition function of a random polymer model with gamma-distributed weights recently introduced by Seppäläinen. We show that the partition function of this random polymer can be represented within the framework of the geometric RSK correspondence and consequently its law can be expressed in terms of Whittaker functions. This leads to a representation of the law of the partition function which is amenable to asymptotic analysis. In this model, the partition function plays a role analogous to the smallest eigenvalue in the Laguerre unitary ensemble of random matrix theory.

We study the contact process on the configuration model with a power law degree distribution, when the exponent is smaller than or equal to two.

We prove that the extinction time grows exponentially fast with the size of the graph and prove two metastability results. First the extinction time divided by its mean converges in distribution toward

an exponential random variable with mean one, when the size of the graph tends to infinity. Moreover, the density of infected sites taken at exponential times converges in probability to a constant. This extends previous results in the case of an exponent larger than $2$ obtained previously.

We study the asymptotic behavior of the diameter or maximum interpoint distance of a cloud of i.i.d. d-dimensional random vectors when the number of points in the cloud tends to infinity. This is a non standard extreme value problem since the diameter is a max $U$-statistic, hence the maximum of dependent random variables. Therefore, the limiting distributions may not be extreme value distributions. We obtain exhaustive results for the Euclidean diameter of a cloud of elliptical vectors whose Euclidean norm is in the domain of attraction for the maximum of the Gumbel distribution. We also obtain results in other norms for spherical vectors and we give several bi-dimensional generalizations. The main idea behind our results and their proofs is a specific property of random vectors whose norm is in the domain of attraction of the Gumbel distribution: the localization into subspaces of low dimension of vectors with a large norm.

It is well-known that the maximal particle in a branching Brownian motion sits near $\sqrt2 t - \frac{3}{2\sqrt2}\log t$ at time $t$. One may then ask about the paths of particles near the frontier: how close can they stay to this critical curve? Two different approaches to this question have been developed. We improve upon the best-known bounds in each case, revealing new qualitative features including marked differences between the two approaches.

Consider a finite system of competing Brownian particles. They move as Brownian motions with drift and diffusion coefficients depending on their ranks. This includes the case of asymmetric collisions, when the local time of any collision is distributed unevenly between the two colliding particles, see Karatzas, Pal and Shkolnikov (2012). A triple collision occurs if three particles occupy the same site at a given moment. This is sometimes an undesirable phenomenon. Continuing the work of Ichiba, Karatzas and Shkolnikov (2013), we find necessary and sufficient condition for absense of triple collisions. We also prove sufficient conditions for absense of quadruple collisions, of quintuple collisions, and so on. Our method is reduction to reflected Brownian motion in the positive multidimesnional orthant hitting non-smooth parts of the boundary and, more generally, edges of the boundary of certain low dimension.

In a first part, we prove a Lyapunov-type criterion for the $\xi_1$-positive recurrence of absorbed birth and death processes and provide new results on the domain of attraction of the minimal quasi-stationary distribution. In a second part, we study the ergodicity and the convergence of a Fleming-Viot type particle system whose particles evolve independently as a birth and death process and jump on each others when they hit 0. Our main result is that the sequence of empirical stationary distributions of the particle system converges to the minimal quasi-stationary distribution of the birth and death process.

We consider a variant of the radial spanning tree introduced by Baccelli and Bordenave. Like the original model, our model is a tree rooted at the origin, built on the realization of a planar Poisson point process. Unlike it, the paths of our model have independent jumps. We show that locally our diffusively rescaled tree, seen as the collection of the paths connecting its sites to the root, converges in distribution to the Brownian Bridge Web, which is roughly speaking a collection of coalescing Brownian bridges starting from all the points of a planar strip perpendicular to the time axis, and ending at the origin.

The subcritical contact process seen from the rightmost infected site has no invariant measures. We prove that nevertheless it converges in distribution to a quasi-stationary measure supported on finite configurations.

We consider simple random walks on Delaunay triangulations generated by point processes in $\mathbb{R}^d$. Under suitable assumptions on the point processes, we show that the random walk satisfies an almost sure (or quenched) invariance principle. This invariance principle holds for point processes which have clustering or repulsiveness properties including Poisson point processes, Matérn cluster and Matérn hardcore processes. The method relies on the decomposition of the process into a martingale part and a corrector which is proved to be negligible at the diffusive scale.

Let $L$ be a convex cone of real random variables on the probability space $(\Omega,\mathcal{A},P_0)$. The existence of a probability $P$ on $\mathcal{A}$ such that $\begin{equation*} P \sim P_0,\quad E_P\left|X\right| < \infty\,\textrm{and} E_P(X)\leq 0\, \mathrm{for all} X \in L \end{equation*}$ is investigated. Two results are provided. In the first, $P$ is a finitely additive probability, while $P$ is $\sigma$-additive in the second. If $L$ is a linear space then $-X\in L$ whenever $X \in L$, so that $E_P(X)\leq 0$ turns into $E_P(X)=0$. Hence, the results apply to various significant frameworks, including equivalent martingale measures and equivalent probability measures with given marginals.

In the compulsive gambler process there is a finite set of agents who meet pairwise at random times ($i$ and $j$ meet at times of a rate-$\nu_{ij}$ Poisson process) and, upon meeting, play an instantaneous fair game in which one wins the other's money. We introduce this process and describe some of its basic properties. Some properties are rather obvious (martingale structure; comparison with Kingman coalescent) while others are more subtle (an "exchangeable over the money elements" property, and a construction reminiscent of the Donnelly-Kurtz look-down construction). Several directions for possible future research are described. One - where agents meet neighbors in a sparse graph - is studied here, and another - a continuous-space extension called the metric coalescent - is studied in Lanoue (2014).

We derive two-sided bounds for expected values of suprema of canonical processes based on random variables with moments growing regularly. We also discuss a Sudakov-type minoration principle for canonical processes.

We show that an algorithmic construction of sequences of recursive trees leads to a direct proof of the convergence of random recursive trees in an associated Doob-Martin compactification; it also gives a representation of the limit in terms of the input sequence of the algorithm. We further show that this approach can be used to obtain strong limit theorems for various tree functionals, such as path length or the Wiener index.

This paper describes two explicit couplings of standard Brownian motions $B$ and $V$, which naturally extend the mirror coupling and the synchronous coupling and respectively maximise and minimise (uniformly over all time horizons) the coupling time and the tracking error of two regime-switching martingales. The generalised mirror coupling minimizes the coupling time of the two martingales while simultaneously maximising the tracking error for all time horizons. The generalised synchronous coupling maximises the coupling time and minimises the tracking error over all co-adapted couplings. The proofs are based on the Bellman principle. We give counterexamples to the conjectured optimality of the two couplings amongst a wider classes of stochastic integrals.

This article gives an account on various aspects of stochastic calculus in the plane. Specifically, our aim is 3-fold: (i) Derive a pathwise change of variable formula for a path $x:[0,1]^{2}\to\mathbb{R}$ satisfying some Hölder regularity conditions with a Hölder exponent greater than $1/3$. (ii) Get some Skorohod change of variable formulas for a large class of Gaussian processes defined on $[0,1]^{2}$. (iii) Compare the bidimensional integrals obtained with those two methods, computing explicit correction terms whenever possible. As a byproduct, we also give explicit forms of corrections in the respective change of variable formulas.

Let $M_{l,n}$ be the number of blocks with frequency $l$ in the exchangeable random partition induced by a sample of size $n$ from the Ewens-Pitman sampling model. In this paper we show that, as $n$ tends to infinity, $n^{-1}M_{l,n}$ satisfies a large deviation principle and we characterize the corresponding rate function. A conditional counterpart of this large deviation principle is also presented. Specifically, given an initial observed sample of size $n$ from the Ewens-Pitman sampling model, we consider an additional unobserved sample of size $m$ thus giving rise to an enlarged sample of size $n+m$. Then, for any fixed $n$ and as $m$ tends to infinity, we establish a large deviation principle for the conditional number of blocks with frequency $l$ in the enlarged sample, given the initial sample. Interestingly this conditional large deviation principle coincides with the large deviation principle for $M_{l,n}$, namely there is no long lasting impact of the given initial sample to the large deviations. Potential applications of our conditional large deviation principle are thoroughly discussed in the context of Bayesian nonparametric inference for species sampling problems.

We consider the Poisson cylinder model in $d$-dimensional hyperbolic space. We show that in contrast to the Euclidean case, there is a phase transition in the connectivity of the collection of cylinders as the intensity parameter varies. We also show that for any non-trivial intensity, the diameter of the collection of cylinders is infinite.

In this paper we consider a branching particle system consisting of particles moving according to an Ornstein-Uhlenbeck process in $\mathbb{R}d$ and undergoing binary, supercritical branching with a constant rate $\lambda > 0$. This system is known to fulfil a law of large numbers (under exponential scaling). In the paper we prove the corresponding central limit theorem. The limit and the CLT normalization fall into three qualitatively different classes. In what we call the small branching rate case the situation resembles the classical one. The weak limit is Gaussian and normalization is the square root of the size of the system. In the critical case the limit is still Gaussian, but the normalization requires an additional term. Finally, when branching has a large rate the situation is completely different. The limit is no longer Gaussian, the normalization is substantially larger than the classical one and the convergence holds in probability. We also prove that the spatial fluctuations are asymptotically independent of the fluctuations of the total number of particles (which is a Galton-Watson process).

We consider supercritical Bernoulli bond percolation on a large $b$-ary tree, in the sense that with high probability, there exists a giant cluster. We show that the size of the giant cluster has non-gaussian fluctuations, which extends a result due to Schweinsberg in the case of random recursive trees. Using ideas in the recent work of Bertoin and Uribe Bravo, the approach developed in this work relies on the analysis of the sub-population with ancestral type in a system of branching processes with rare mutations, which may be of independent interest. This also allows us to establish the analogous result for scale-free trees.

Excited deterministic walk in a random environment is a non-Markov integer-valued process $(X_n)_{n=0}^{\infty}$, whose jump at time $n$ depends on the number of visits to the site $X_n$. The environment can be understood as stacks of cookies on each site of $\mathbb Z$. Once all cookies are consumed at a given site, every subsequent visit will result in a walk taking a step according to the direction prescribed by the last consumed cookie. If each site has exactly one cookie, then the walk ends in a loop if it ever visits the same site twice. If the number of cookies per site is increased to two, the walk can visit a site $x$ arbitrarily many times before getting stuck in a loop, which may or may not contain $x$. Nevertheless, we establish monotonicity results on the environment that imply large deviations.

We consider standard $\Lambda$-coalescents (or coalescents with multiple collisions) with a non-trivial "Kingman part". That is, the driving measure $\Lambda$ has an atom at $0; \Lambda(\{0\})= c > 0$. It is known that all such coalescents come down from infinity. Moreover, the number of blocks $N_t$ is asymptotic to $v(t) = 2/(ct)$ as $t\to 0$. In the present paper we investigate the second-order asymptotics of $N_t$ in the functional sense at small times. This complements our earlier results on the fluctuations of the number of blocks for a class of regular $\Lambda$-coalescents without the Kingman part. In the present setting it turns out that the Kingman part dominates and the limit process is a Gaussian diffusion, as opposed to the stable limit in our previous work.

Following a hedging based approach to model free financial mathematics, we prove that it should be possible to make an arbitrarily large profit by investing in those one-dimensional paths which do not possess local times. The local time is constructed from discrete approximations, and it is shown that it is $\alpha$-Hölder continuous for all $\alpha < 1/2$. Additionally, we provide various generalizations of Föllmer's pathwise Itô formula.

We consider the Gaussian free field on $Z^d$, $d\geq3$, and prove that the critical density for percolation of its level sets behaves like $1/d^{1+o(1)}$ as $d$ tends to infinity. Our proof gives the principal asymptotic behavior of the corresponding critical level $h_*(d)$. Moreover, it shows that a related parameter $h_{**}(d)$ introduced by Rodriguez and Sznitman in [23] is in fact asymptotically equivalent to $h_*(d)$.

Small-space and large-time estimates and asymptotic expansion of the distribution function and (the derivatives of) the density function of hitting times of points for symmetric Lévy processes are studied. The Lévy measure is assumed to have completely monotone density function, and a scaling-type condition $\mathrm{inf} \xi \Psi"(\xi) / \Psi'(\xi) > 0$ is imposed on the Lévy-Khintchine exponent $\Psi$. Proofs are based on generalised eigenfunction expansion for processes killed upon hitting the origin.

We consider the simple random walk on $\mathbb{Z}^d$ evolving in a random i.i.d. potential taking values in $[0,+\infty)$. The potential is not assumed integrable, and can be rescaled by a multiplicative factor $\lambda > 0$. Completing the work started in a companion paper, we give the asymptotic behaviour of the Lyapunov exponents for $d \ge 3$, both annealed and quenched, as the scale parameter $\lambda$ tends to zero.

Two discretizations of a novel class of Markovian backward stochastic differential equations (BSDEs) are studied. The first is the classical Euler scheme which approximates a projection of the processes $Z$, and the second a novel scheme based on Malliavin weights which approximates the mariginals of the process $;Z$ directly. Extending the representation theorem of Ma and Zhang leads to advanced a priori estimates and stability results for this class of BSDEs. These estimates are then used to obtain competitive convergence rates for both schemes with respect to the number of points in the time-grid. The class of BSDEs considered includes Lipschitz BSDEs with fractionally smooth terminal condition as well as quadratic BSDEs with bounded, Hölder continuous terminal condition.

For a continuous function $f \in \mathcal{C}([0,1])$, define the Vervaat transform $V(f)(t):=f(\tau(f)+t \mod1)+f(1)1_{\{t+\tau(f) \geq 1\}}-f(\tau(f))$, where $\tau(f)$ corresponds to the first time at which the minimum of $f$ is attained. Motivated by recent study of quantile transforms of random walks and Brownian motion, we investigate the Vervaat transform of Brownian motion and Brownian bridges with arbitrary endpoints. When the two endpoints of the bridge are not the same, the Vervaat transform is not Markovian. We describe its distribution by path decomposition and study its semi-martingale property. The same study is done for the Vervaat transform of unconditioned Brownian motion, the expectation and variance of which are also derived.

Given a sequence T_1, T_2, ... of random d-by-d matrices with nonnegative entries, suppose there is a random vector X with nonnegative entries, such that the sum T_1 X_1 + T_2 X_2 +... has the same law as X, with X_1, X_2, ... being i.i.d.copies of X, independent of T_1, T_2, ... Then (the law of) X is called a fixed point of the multivariate smoothing transform. Similar to the well-studied one-dimensional case d=1, a function m is introduced, such that the existence of $\alpha \in (0,1]$ with $m(\alpha)=1$ and $m'(\alpha) \le 0$ guarantees the existence of nontrivial fixed points. We prove the uniqueness of fixed points in the critical case $m'(\alpha)=0$ and describe their tail behavior. This complements recent results for the non-critical multivariate case. Moreover, we introduce the multivariate analogue of the derivative martingale and prove its convergence to a non-trivial limit.

We modify Talagrand's generic chaining method to obtain upper bounds for all p-th moments of the supremum of a stochastic process. These bounds lead to an estimate for the upper tail of the supremum with optimal deviation parameters. We apply our procedure to improve and extend some known deviation inequalities for suprema of unbounded empirical processes and chaos processes. As an application we give a significantly simplified proof of the restricted isometry property of the subsampled discrete Fourier transform.

In this article, we consider the stochastic wave and heat equations on $\mathbb{R}$ with non-vanishing initial conditions, driven by a Gaussian noise which is white in time and behaves in space like a fractional Brownian motion of index $H$, with $1/4 < H < 1/2$. We assume that the diffusion coefficient is given by an affine function $\sigma(x)=ax+b$, and the initial value functions are bounded and Hölder continuous of order $H$. We prove the existence and uniqueness of the mild solution for both equations. We show that the solution is $L^{2}(\Omega)$-continuous and its $p$-th moments are uniformly bounded, for any $p \geq 2$.

This paper studies the stochastic heat equation with multiplicative noises of the form uW, where W is a mean zero Gaussian noise and the differential element uW is interpreted both in the sense of Skorohod and Stratonovich. The existence and uniqueness of the solution are studied for noises with general time and spatial covariance structure. Feynman-Kac formulas for the solutions and for the moments of the solutions are obtained under general and different conditions. These formulas are applied to obtain the Hölder continuity of the solutions. They are also applied to obtain the intermittency bounds for the moments of the solutions.

We consider a transient diffusion in a $(-\kappa/2)$-drifted Brownian potential $W_{\kappa}$ with $0 < \kappa < 1$. We prove its localization at time $t$ in the neighborhood of some random points depending only on the environment, which are the positive $h_t$-minima of the environment, for $h_t$ a bit smaller than $\log t$.We also prove an Aging phenomenon for the diffusion, a renewal theorem for the hitting time of the farthest visited valley, and provide a central limit theorem for the number of valleys visited up to time $t$. The proof relies on adecomposition of the trajectory of $W_{\kappa}$ in the neighborhood of$h_t$-minima, with the help of results of A. Faggionato, and on a precise analysis of exponential functionals of $W_{\kappa}$ and of $W_{\kappa}$ Doob-conditioned to stay positive.;

We study the performance of Empirical Risk Minimization in both noisy and noiseless phase retrieval problems, indexed by subsets of $\mathbb{R}^n$ and relative to subgaussian sampling; that is, when the given data is $y_i=\left a_i,x_0\right^2+w_i$ for a subgaussian random vector $a$, independent noise $w$ and a fixed but unknown $x_0$ that belongs to a given subset of $\mathbb{R}^n$.

We show that ERM produces $\hat{x}$ whose Euclidean distance to either $x_0$ or $-x_0$ depends on the gaussian mean-width of the indexing set and on the signal-to-noise ratio of the problem. The bound coincides with the one for linear regression when $\|x_0\|_2$ is of the order of a constant. In addition, we obtain a minimax lower bound for the problem and identify sets for which ERM is a minimax procedure. As examples, we study the class of $d$-sparse vectors in $\mathbb{R}^n$ and the unit ball in $\ell_1^n$.

Widely used models in genetics include the Wright-Fisher diffusion and its moment dual, Kingman's coalescent. Each has a multilocus extension but under neither extension is the sampling distribution available in closed-form, and their computation is extremely difficult. In this paper we derive two new multilocus population genetic models, one a diffusion and the other a coalescent process, which are much simpler than the standard models, but which capture their key properties for large recombination rates. The diffusion model is based on a central limit theorem for density dependent population processes, and we show that the sampling distribution is a linear combination of moments of Gaussian distributions and hence available in closed-form. The coalescent process is based on a probabilistic coupling of the ancestral recombination graph to a simpler genealogical process which exposes the leading dynamics of the former. We further demonstrate that when we consider the sampling distribution as an asymptotic expansion in inverse powers of the recombination parameter, the sampling distributions of the new models agree with the standard ones up to the first two orders.

In a previous paper, the authors introduced an approach to prove that the statistics of the extremes of a log-correlated Gaussian field converge to a Poisson-Dirichlet variable at the level of the Gibbs measure at low temperature and under suitable test functions.The method is based on showing that the model admits a one-step replica symmetry breaking in spin glass terminology.This implies Poisson-Dirichlet statistics by general spin glass arguments.In this note, this approach is used to prove Poisson-Dirichlet statistics for the two-dimensional discrete Gaussian free field, where boundary effects demand a more delicate analysis.

This paper introduces a matrix analog of the Bessel processes, taking values in the closed set $E$ of real square matrices with nonnegative determinant. They are related to the well-known Wishart processes in a simple way: the latter are obtained from the former via the map $x\mapsto x^\top x$. The main focus is on existence and uniqueness via the theory of Dirichlet forms. This leads us to develop new results of potential theoretic nature concerning the space of real square matrices. Specifically, the function $w(x)=|\det x|^\alpha$ is a weight function in the Muckenhoupt $A_p$ class for $-1 < \alpha\le 0$ ($p=1$) and $-1 < \alpha < p-1$ ($p > 1$). The set of matrices of co-rank at least two has zero capacity with respect to the measure $m(dx)=|\det x|^\alpha dx$ if $\alpha >-1$, and if $\alpha\ge 1$ this even holds for the set of all singular matrices. As a consequence we obtain density results for Sobolev spaces over (the interior of) $E$ with Neumann boundary conditions. The highly non-convex, non-Lipschitz structure of the state space is dealt with using a combination of geometric and algebraic methods.

This paper is concerned with various aspects of the Slepian process $(B_{t+1} - B_t, t \ge 0)$ derived from a one-dimensional Brownian motion $(B_t, t \ge 0 )$. In particular, we offer an analysis of the local structure of the Slepian zero set $\{t : B_{t+1} = B_t \}$, including a path decomposition of the Slepian process for $0 \le t \le 1$. We also establish the existence of a random time $T$ such that $T$ falls in the the Slepian zero set almost surely and the process $(B_{T+u} - B_T, 0 \le u \le 1)$ is standard Brownian bridge.

We consider the slow movement of randomly biased random walk $(X_n)$ on a supercritical Galton-Watson tree, and are interested in the sites on the tree that are most visited by the biased random walk. Our main result implies tightness of the distributions of the most visited sites under the annealed measure. This is in contrast with the one-dimensional case, and provides, to the best of our knowledge, the first non-trivial example of null recurrent random walk whose most visited sites are not transient, a question originally raised by Erdős and Révész for simple symmetric random walk on the line.

We provide some on-off type criteria for recurrence of regime-switching diffusion processes using the theory of M-matrix, the Perron-Frobenius theorem. State-independent and state-dependent regime-switching diffusion processes in a finite space and an infinite countable space are both studied. Especially, we put forward a finite partition method to deal with switching process in an infinite countable space. As an application, we study the recurrence of regime-switching Ornstein-Uhlenbeck process, and provide an on-off type criterion for a kind of nonlinear regime-switching diffusion processes.

We introduce a Voter Model variant, inspired by social evolution of musical preferences. In our model, agents have preferences over a set of songs and upon meeting update their own preferences incrementally towards those of the other agents they meet. Using the spectral gap of an associated Markov chain, we give a geometry dependent result on the asymptotic consensus time of the model.

In this paper, we pursue the study of second order BSDEs with jumps (2BSDEJs for short) started in an accompanying paper. We prove existence of these equations by a direct method, thus providing complete wellposedness for 2BSDEJs. These equations are a natural candidate for the probabilistic interpretation of some fully non-linear partial integro-differential equations, which is the point of the second part of this work. We prove a non-linear Feynman-Kac formula and show that solutions to 2BSDEJs provide viscosity solutions of the associated PIDEs.

In this article, we prove the existence of bounded solutions of quadratic backward SDEs with jumps, that is to say for which the generator has quadratic growth in the variables (z; u). From a technical point of view, we use a direct fixed point approach as in Tevzadze [38], which allows us to obtain existence and uniqueness of a solution when the terminal condition is small enough. Then, thanks to a well-chosen splitting, we recover an existence result for general bounded solution. Under additional assumptions, we can obtain stability results and a comparison theorem, which as usual implies uniqueness.

Let $E$ be a complete, separable metric space and $A$ be an operator on $C_b(E)$. We give an abstract definition of viscosity sub/supersolution of the resolvent equation $\lambda u-Au=h$ and show that, if the comparison principle holds, then the martingale problem for $A$ has a unique solution. Our proofs work also under two alternative definitions of viscosity sub/supersolution which might be useful, in particular, in infinite dimensional spaces, for instance to study measure-valued processes.

We prove the analogous result for stochastic processes that must satisfy boundary conditions, modeled as solutions of constrained martingale problems. In the case of reflecting diffusions in $D\subset {\bf R}^d$, our assumptions allow $D$ to be nonsmooth and the direction of reflection to be degenerate.

Two examples are presented: A diffusion with degenerate oblique direction of reflection and a class of jump diffusion processes with infinite variation jump component and possibly degenerate diffusion matrix.

In this article, we study a branching random walk in an environment which depends on the time. This time-inhomogeneous environment consists of a sequence of macroscopic time intervals, in each of which the law of reproduction remains constant. We prove that the asymptotic behaviour of the maximal displacement in this process consists of a first ballistic order, given by the solution of an optimization problem under constraints, a negative logarithmic correction, plus stochastically bounded fluctuations.

We consider an infinite system of Brownian motions which interact through a given Brownian motion being reflected from its left neighbor. Earlier we studied this system for deterministic periodic initial configurations. In this contribution we consider initial configurations distributed according to a Poisson point process with constant intensity, which makes the process space-time stationary. We prove convergence to the Airy process for stationary the case. As a byproduct we obtain a novel representation of the finite-dimensional distributions of this process. Our method differs from the one used for the TASEP and the KPZ equation by removing the initial step only after the limit $t\to\infty$. This leads to a new universal cross-over process.

In this paper, we prove that the time supremum of the Wasserstein distance between the time-marginals of a uniformly elliptic multidimensional diffusion with coefficients bounded together with their derivatives up to the order $2$ in the spatial variables and Hölder continuous with exponent $\gamma$ with respect to the time variable and its Euler scheme with $N$ uniform time-steps is smaller than $C(1+\mathbf{1}_{\gamma=1} \sqrt{\ln(N)})N^{-\gamma}$. To do so, we use the theory of optimal transport. More precisely, we investigate how to apply the theory by Ambrosio et al. to compute the time derivative of the Wasserstein distance between the time-marginals. We deduce a stability inequality for the Wasserstein distance which finally leads to the desired estimation.

We investigate the disordered copolymer and pinning models, in the case of a correlated Gaussian environment with correlations, and when the return distribution of the underlying renewal process has a polynomial tail. As far as the copolymer model is concerned, we prove disorder relevance both in terms of critical points and critical exponents, in the case of non-negative correlations. When some of the correlations are negative, even the annealed model becomes non-trivial. Moreover, when the return distribution has a finite mean, we are able to compute the weak coupling limit of the critical curves for both models, with no restriction on the correlations other than summability. This generalizes the result of Berger,Caravennale, Poisat, Sun and Zygouras to the correlated case. Interestingly, in the copolymer model, the weak coupling limit of the critical curve turns out to be the maximum of two quantities: one generalizing the limit found in the IID case, the other one generalizing the so-called Monthus bound.

We introduce a method for studying monotonicity of the speed of excited random walks in high dimensions, based on a formula for the speed obtained via cut-times and Girsanov's transform. While the method gives rise to similar results as have been or can be obtained via the expansion method of van der Hofstad and Holmes, it may be more palatable to a general probabilistic audience. We also revisit the law of large numbers for stationary cookie environments. In particular, we introduce a new notion of $\mathcal{e}_1 -$ exchangeable cookie environment and prove the law of large numbers for this case.

We give criteria on the existence of a so-called mark function in the context of marked metric measure spaces (mmm-spaces). If an mmm-space admits a mark function, we call it functionally-marked metric measure space (fmm-space). This is not a closed property in the usual marked Gromov-weak topology, and thus we put particular emphasis on the question under which conditions it carries over to a limit. We obtain criteria for deterministic mmm-spaces as well as random mmm-spaces and mmm-space-valued processes. As an example, our criteria are applied to prove that the tree-valued Fleming-Viot dynamics with mutation and selection from previous works admits a mark function at all times, almost surely. Thereby, we fill a gap in a former proof of this fact, which used a wrong criterion. Furthermore, the subspace of fmm-spaces, which is dense and not closed, is investigated in detail. We show that there exists a metric that induces the marked Gromov-weak topology on this subspace and is complete. Therefore, the space of fmm-spaces is a Polish space. We also construct a decomposition into closed sets which are related to the case of uniformly equicontinuous mark functions.

We consider weighted graphs satisfying sub-Gaussian estimate for the natural random walk. On such graphs, we study symmetric Markov chains with heavy tailed jumps. We establish a threshold behavior of such Markov chains when the index governing the tail heaviness (or jump index) equals the escape time exponent (or walk dimension) of the sub-Gaussian estimate. In a certain sense, this generalizes the classical threshold corresponding to the second moment condition.

;We establish a general sufficient condition for a sequence of Galton-Watson branching processes in varying environments to converge weakly. This condition extends previous results by allowing offspring distributions to have infinite variance. Our assumptions are stated in terms of pointwise convergence of a triplet of two real-valued functions and a measure. The limiting process is characterized by a backwards integro-differential equation satisfied by its Laplace exponent, which generalizes the branching equation satisfied by continuous state branching processes. Several examples are discussed, namely branching processes in random environment, Feller diffusion in varying environments and branching processes with catastrophes.

Earlier work by Diaconis and Saloff-Coste gives a spectral criterion for a maximum separation cutoff to occur for birth and death chains. Ding, Lubetzky and Peres gave a related criterion for a maximum total variation cutoff to occur in the same setting. Here, we provide complementary results which allow us to compute the cutoff times and windows in a variety of examples.

We study the limiting object of a sequence of Markov chains analogous to the limits of graphs, hypergraphs, and other objects which have been studied. Following a suggestion of Aldous, we assign to a convergent sequence of finite Markov chains with bounded mixing times a unique limit object: an infinite Markov chain with a measurable state space. The limits of the Markov chains we consider have discrete spectra, which makes the limit theory simpler than the general graph case, and illustrates how the discrete spectrum setting (sometimes called "random-free" or "product measurable") is simpler than the general case.

We derive the limiting distribution of the barycenter $b_n$ of an i.i.d. sample of $n$ random points on a planar cone with angular spread larger than $2\pi$. There are three mutually exclusive possibilities: (i) (fully sticky case) after a finite random time the barycenter is almost surely at the origin; (ii) (partly sticky case) the limiting distribution of $\sqrt{n} b_n$ comprises a point mass at the origin, an open sector of a Gaussian, and the projection of a Gaussian to the sector's bounding rays; or (iii) (nonsticky case) the barycenter stays away from the origin and the renormalized fluctuations have a fully supported limit distribution-usually Gaussian but not always. We conclude with an alternative, topological definition of stickiness that generalizes readily to measures on general metric spaces.

We prove a version of McDiarmid’s bounded differences inequality for Markov chains, with constants proportional to the mixing time of the chain. We also show variance bounds and Bernstein-type inequalities for empirical averages of Markov chains. In the case of non-reversible chains, we introduce a new quantity called the “pseudo spectral gap”, and show that it plays a similar role for non-reversible chains as the spectral gap plays for reversible chains. Our techniques for proving these results are based on a coupling construction of Katalin Marton, and on spectral techniques due to Pascal Lezaud. The pseudo spectral gap generalises the multiplicative reversiblication approach of Jim Fill.

The limits of scaled relative entropies between probability distributions associated with $N$-particle weakly interacting Markov processes are considered. The convergence of such scaled relative entropies is established in various settings. The analysis is motivated by the role relative entropy plays as a Lyapunov function for the (linear) Kolmogorov forward equation associated with an ergodic Markov process, and Lyapunov function properties of these scaling limits with respect to nonlinear finite-state Markov processes are studied in the companion paper

The focus of this work is on local stability of a class of nonlinear ordinary differential equations (ODE) that describe limits of empirical measures associated with finite-state exchangeable weakly interacting N -particle systems. Local Lyapunov functions are identified for several classes of such ODE, including those associated with systems with slow adaptation and Gibbs systems. Using previous results and large deviations heuristics, a partial differential equation (PDE) associated with the nonlinear ODE is introduced and it is shown that positive definite subsolutions of this PDE serve as local Lyapunov functions for the ODE. This PDE characterization is used to construct explicit Lyapunov functions for a broad class of models called locally Gibbs systems. This class of models is significantly larger than the family of Gibbs systems and several examples of such systems are presented, including models with nearest neighbor jumps and models with simultaneous jumps that arise in applications.

Suppose $X$ is a time-homogeneous diffusion on an interval $I^X \subseteq {\mathbb R}$ and let $\mu$ be a probability measure on $I^X$. Then $\tau$ is a solution of the Skorokhod embedding problem (SEP) for $\mu$ in $X$ if $\tau$ is a stopping time and $X_\tau \sim \mu$. There are well-known conditions which determine whether there exists a solution of the SEP for $\mu$ in $X$. We give necessary and sufficient conditions for there to exist an integrable solution. Further, if there exists a solution ofthe SEP then there exists a minimal solution. We show that every minimal solution of the SEP has the same first moment. When $X$ is Brownian motion, there exists an integrable embedding of $\mu$ if and only if $\mu$ is centred and in $L^2$. Further,every integrable embedding is minimal. When $X$ is a general time-homogeneous diffusion the situation is more subtle. The case with drift can be reduced to the local martingale case by a change of scale. If $Y$ is a diffusion in natural scale, and if the target law is centred, then as in the Brownian case, there is an integrable embedding if the target law satisfies an integral condition. However, unlike in the Brownian case, there exist integrable embeddings of target laws which are not centred. Further, there exist integrable embeddings which are not minimal. Instead, if there exists an integrable embedding, then the set of minimal embeddings is the set of embeddings such that the mean equals a certain quantity, which we identify.

In this work we connect the theory of symmetric Dirichlet forms and direct stochastic calculus to obtain strong existence and pathwise uniqueness for Brownian motion that is perturbed by a series of constant multiples of local times at a sequence of points that has exactly one accumulation point in $\mathbb{R}$. The considered process is identified as special distorted Brownian motion $X$ in dimension one and is studied thoroughly. Besides strong uniqueness, we present necessary and sufficient conditions for non-explosion, recurrence and positive recurrence as well as for $X$ to be semimartingale and possible applications to advection-diffusion in layered media.

Double (or parity conserving) branching annihilating random walk, introduced by Sudbury in '90, is a one-dimensional non-attractive particle system in which positive and negative particles perform nearest neighbor hopping, produce two offsprings to neighboring lattice points and annihilate when they meet. Given an odd number of initial particles, positive recurrence as seen from the leftmost particle position was first proved by Belitsky, Ferrari, Menshikov and Popov in '01 and, subsequently in a much more general setup, in the article by Sturm and Swart (Tightness of voter model interfaces) in '08. These results assume that jump rates of the various moves do not depend on the configuration of the particles not involved in these moves. The present article deals with the case when the jump rates are affected by the locations of several particles in the system. Motivation for such models comes from non-attractive interacting particle systems with particle conservation. Under suitable assumptions we establish the existence of the process, and prove that the one-particle state is positive recurrent. We achieve this by arguments similar to those appeared in the previous article by Sturm and Swart. We also extend our results to some cases of long range jumps, when branching can also occur to non-neighboring sites. We outline and discuss several particular examples of models where our results apply.

Motivated by credit risk modelling, we consider a type of default times whose probability law can have atoms, where standard intensity and density hypotheses in the enlargement of filtrations are not satisfied. We propose a generalized density approach in order to treat such random times in the framework of progressive enlargement of filtrations. We determine the compensator process of the random time and study the martingale and semimartingale processes in the enlarged filtration which are important for the change of probability measures and the evaluation of credit derivatives. The generalized density approach can also be applied to model simultaneous default events in the multi-default setting.

A stochastic particle model for fragmentation process is considered. Evolution of the system of particles is described by a stochastic process on a space of discrete measures on a Polish space. A phenomenon of shattering into dust is studied and some criteria for mass conservation and loss of mass in our model are proven.

We discuss chains of interacting Brownian motions. Their time reversal invariance is broken because of asymmetry in the interaction strength between left and right neighbor. In the limit of a very steep and short range potential one arrives at Brownian motions with oblique reflections. For this model we prove a Bethe ansatz formula for the transition probability and self-duality. In case of half-Poisson initial data, duality is used to arrive at a Fredholm determinant for the generating function of the number of particles to the left of some reference point at any time $t &gt; 0$. A formal asymptotics for this determinant establishes the link to the Kardar-Parisi-Zhang universality class.

We study the almost sure (a.s.) behaviour of a Lévy process $(X_t)_{t\ge 0}$ on $\mathbb{R}$ with extreme values removed, giving necessary and sufficient conditions for the a.s. convergence as $t\to0$ of normed and centered versions of "trimmed" processes, in which the $r$ largest positive jumps or the $r$ largest jumps in modulus of $X$ up to time $t$ are subtracted from it. Integral criteria in terms of the canonical measure of $X$ are given for the required convergences, under natural conditions on the norming functions. Random walk results of Mori (1976, 1977) and Lévy process results of Shtatland (1965) and Rogozin (1968) are thereby generalised. Another application is to characterise the relative stability at 0 of the trimmed processes, in probability and almost surely.<br />

This paper is concerned with the existence, shape and dynamical stability of infinite-energy equilibria for a class of spatially homogeneous kinetic equations in space dimensions $d\ge2$. Our results cover in particular Bobylev's model for inelastic Maxwell molecules. First, we show under certain conditions on the collision kernel, that there exists an index $\alpha\in(0,2)$ such that the equation possesses a nontrivial stationary solution, which is a scale mixture of radially symmetric $\alpha$-stable laws. We also characterize the mixing distribution as the fixed point of a smoothing transformation. Second, we prove that any transient solution that emerges from the NDA of some (not necessarily radial symmetric) $\alpha$-stable distribution converges to an equilibrium. The key element of the convergence proof is an application of the central limit theorem to a representation of the transient solution as a weighted sum of projections of randomly rotated i.i.d. random vectors.

We examine a new path transform on 1-dimensional simple random walks and Brownian motion, the quantile transform. This transformation relates to identities in fluctuation theory due to Wendel, Port, Dassios and others, and to discrete and Brownian versions of Tanaka’s formula. For an $n$-step random walk, the quantile transform reorders increments according to the value of the walk at the start of each increment. We describe the distribution of the quantile transform of a simple random walk of $n$ steps, using a bijection to characterize the number of pre-images of each possible transformed path. We deduce, both for simple random walks and for Brownian motion, that the quantile transform has the same distribution as Vervaat’s transform. For Brownian motion, the quantile transforms of the embedded simple random walks converge to a time change of the local time profile. We characterize the distribution of the local time profile, giving rise to an identity that generalizes a variant of Jeulin’s description of the local time profile of a Brownian bridge or excursion.

In this paper we establish local estimates for the first passage time of a subordinator under the assumption that it belongs to the Feller class, either at zero or infinity, having as a particular case the subordinators which are in the domain of attraction of a stable distribution, either at zero or infinity. To derive these results we first obtain uniform local estimates for the one dimensional distribution of such a subordinator, which sharpen those obtained by Jain and Pruitt. In the particular case of a subordinator in the domain of attraction of a stable distribution our results are the analogue of the results obtained by the authors for non-monotone Levy processes. For subordinators an approach different to that in [6] is necessary because the excursion techniques are not available and also because typically in the non-monotone case the tail distribution of the first passage time has polynomial decrease, while in the subordinator case it is exponential.

In this paper we prove that the spatially homogeneous Landau equation for Maxwellian moleculescan be represented through the product of two elementary stochastic processes. The first one is the Brownian motion on the group of rotations. The second one is, conditionally on the first one, a Gaussian process. Using this representation, we establish sharp multi-scale upper and lower bounds for the transition density of the Landau equation, the multi-scale structure depending on the shape of the support of the initial condition.

Explicit sufficient conditions on the hypercontractivity are presented for two classes of functional stochastic partial differential equationsdriven by, respectively, non-degenerate and degenerate Gaussian noises. Consequently, these conditions imply that the associated Markov semigroup is $L^2$-compact and exponentially convergent to the stationary distribution in entropy, variance and total variational norm. As the log-Sobolev inequality is invalid under the framework, we apply a criterion presented in a recent paper using Harnack inequality, coupling property and Gaussian concentration property of the stationary distribution. To verify the concentration property, we prove a Fernique type inequality for infinite-dimensional Gaussian processes which might be interesting by itself.

The genealogy at a single locus of a constant size $N$ population in equilibrium is given by the well-known Kingman's coalescent. When considering multiple loci under recombination, the ancestral recombination graph encodes the genealogies at all loci in one graph. For a continuous genome $\mathbb G$, we study the tree-valued process $(T^N_u)_{u\in\mathbb{G}}$ of genealogies along the genome in the limit $N\to\infty$. Encoding trees as metric measure spaces, we show convergence to a tree-valued process with cadlag paths. In addition, we study mixing properties of the resulting process for loci which are far apart.

In this paper we study a random walk in a one-dimensional dynamic random environment consisting of a collection of independent particles performing simple symmetric random walks in a Poisson equilibrium with density $\rho \in (0,\infty)$. At each step the random walk performs a nearest-neighbour jump, moving to the right with probability $p_{\circ}$ when it is on a vacant site and probability $p_{\bullet}$ when it is on an occupied site. Assuming that $p_\circ \in (0,1)$ and $p_\bullet \neq \tfrac12$, we show that the position of the random walk satisfies a strong law of large numbers, a functional central limit theorem and a large deviation bound, provided $\rho$ is large enough. The proof is based on the construction of a renewal structure together with a multiscale renormalisation argument.

In this paper we investigate the local risk-minimization approach for a semimartingale financial market where there are restrictions on the available information to agents who can observe at least the asset prices. We characterize the optimal strategy in terms of suitable decompositions of a given contingent claim, with respect to a filtration representing the information level, even in presence of jumps. Finally, we discuss an application to a Markovian framework and show that the computation of the optimal strategy leads to filtering problems under the real-world probability measure and under the minimal martingale measure.

We give a sufficient condition for a random sequence in [0,1] generated by a Psi-process to be equidistributed. The condition is met by the canonical example - the max-2 process - where the $n$th term is whichever of two uniformly placed points falls in the larger gap formed by the previous $n$-1 points. Also, we deduce equidistribution for an interpolation of the min-2 and max-2 processes that is biased towards min-2, as well as more general interpolations. This solves an open problem from Itai Benjamini, Pascal Maillard and Elliot Paquette.

We consider a natural destruction process of an infinite recursive tree by removing each edge after an independent exponential time. The destruction up to time $t$ is encoded by a partition $\Pi(t)$ of $\mathbb{N}$ into blocks of connected vertices. Despite the lack of exchangeability, just like for an exchangeable fragmentation process, the process $\Pi$ is Markovian with transitions determined by a splitting rates measure ${\bf r}$. However, somewhat surprisingly, ${\bf r}$ fails to fulfill the usual integrability condition for the dislocation measure of exchangeable fragmentations. We further observe that a time-dependent normalization enables us to define the weights of the blocks of $\Pi(t)$. We study the process of these weights and point at connections with Ornstein-Uhlenbeck type processes.<br /><br />

We prove an asymptotic shape theorem for first-passage percolation on Cayley graphs of virtually nilpotent groups. By a theorem of Pansu, the asymptotic cone of a finitely generated nilpotent group is isometric to a simply connected nilpotent Lie group equipped with some left invariant Carnot-Caratheodory metric. Our main result is an extension of Pansu's theorem to random metrics, where the edges of the Cayley are i.i.d. random variable with some finite exponential moment. Based on a companion work by the second author, the proof relies on Talagrand's concentration inequality, and on Pansu's theorem. Adapting an argument of Benjamini, Kalai and Schramm, we prove a sublinear estimate on the variance for virtually nilpotent groups which are not virtually isomorphic to Z. We further discuss the asymptotic cones of first-passage percolation on general infinite connected graphs: we prove that the asymptotic cones are a.e. deterministic if and only the volume growth is subexponential.<br />

Consider the dynamics of a particle whose speed satisfies a one-dimensional stochastic differential equation driven by a small symmetric $\alpha$-stable Lévy process in a potential of the form a power function of exponent $\beta+1$. Two cases are studied: the noise could be path continuous, namely a standard Brownian motion, if $\alpha=2$, or pure jump Lévy process, if $\alpha\in(0,2)$. The main goal is to study a scaling limit of the position process with this speed, and one proves that the limit is Brownian in either case. This result is a generalization in some sense of the quadratic potential case studied recently by Hintze and Pavlyukevich.<br />

We investigate the growth of clusters within the forest fire model of Ráth and Tóth [EJP, vol 14, paper no 45]. The model is a continuous-time Markov process, similar to the dynamical Erdős-Rényi random graph but with the addition of so-called fires. A vertex may catch fire at any moment and, when it does so, causes all edges within its connected cluster to burn, meaning that they instantaneously disappear. Each burned edge may later reappear.

We give a precise description of the process $C_t$ of the size of the cluster of a tagged vertex, in the limit as the number of vertices in the model tends to infinity. We show that $C_t$ is an explosive branching process with a time-inhomogeneous offspring distribution and instantaneous return to 1 on each explosion. Additionally, we show that the characteristic curves used to analyse the Smoluchowski-type coagulation equations associated to the model have a probabilistic interpretation in terms of the process $C_t$.

We consider the $q$-Hahn TASEP which is a three-parameter family of discrete time interacting particle systems. The particles jump to the right independently according to a certain $q$-Binomial distribution with parallel updates. It is a generalization of the totally asymmetric simple exclusion process (TASEP) on $\mathbb{Z}$. For step initial condition, we prove that the current fluctuation of $q$-Hahn TASEP at time $\tau$ is of order $\tau^{1/3}$ and asymptotically distributed as the GUE Tracy–Widom distribution. We verify the KPZ scaling theory conjecture for the $q$-Hahn TASEP.

Ehrenborg, Levin and Readdy have introduced a new probabilistic approachto the combinatorics of permutations with fixed set of descents. In this paper we extend this approach by introducing a more general probabilistic model. The study ofthis model yields new estimates on the behavior of a uniform random permutation σhaving a fixed descent set. In particular, we find a positive answer to a conjecture and we show that independently of the shape of the descent set, $σ(i)$ and $σ(j)$ are almost independent when $i − j$ becomes large.

We study the characteristic polynomial of Haar distributed random unitary matrices. We show that after a suitable normalization, as one increases the size of the matrix, powers of the absolute value of the characteristic polynomial as well as powers of the exponential of its argument converge in law to a Gaussian multiplicative chaos measure for small enough real powers. This establishes a connection between random matrix theory and the theory of Gaussian multiplicative chaos.

We prove strong law of large numbers and an annealed invariance principle for a random walk in a one-dimensional dynamic random environment evolving as the simple exclusion process with jump parameter $\gamma$. First we establish that, if the asymptotic velocity of the walker is non-zero in the limiting case "$\gamma = \infty$" where the environment gets fully refreshed between each step, then, for $\gamma$ large enough, the walker still has a non-zero asymptotic velocity in the same direction. Second we establish that if the walker is transient in the limiting case $\gamma = 0$, then, for $\gamma$ small enough but positive, the walker has a non-zero asymptotic velocity in the direction of the transience. These two limiting velocities can sometimes be of opposite sign. In all cases, we show that fluctuations are normal.

We study the corrector equation in stochastic homogenization for a simplified Bernoulli percolation model on $\mathbb{Z}^d$, $d > 2$. The model is obtained from the classical $\{0,1\}$-Bernoulli bond percolation by conditioning all bonds parallel to the first coordinate direction to be open. As a main result we prove (in fact for a slightly more general model) that stationary correctors exist and that all finite moments of the corrector are bounded. This extends a previous result of Gloria & Otto, where uniformly elliptic conductances are treated, to the degenerate case. With regard to the associated random conductance model, we obtain as a side result that the corrector not only grows subinearly, but slower than any polynomial rate. Our argument combines a quantification of ergodicity by means of a Spectral Gap on Glauber dynamics with regularity estimates on the gradient of the elliptic Green's function.

In dense Erdős–Rényi random graphs, we are interested in the events where large numbers of a given subgraphs occur. The mean behaviour of subgraph counts is known, and only recently were the related large deviations results discovered. Consequently, it is natural to ask, what is the probability of an Erdős–Rényi graph containing an excessively large number of a given subgraph? Using the large deviation principle, we study an importance sampling scheme as a method to numerically compute the small probabilities of large triangle counts occurring within Erdős–Rényi graphs. The exponential tilt used in the importance sampling scheme comes from a generalized class of exponential random graphs. Asymptotic optimality, a measure of the efficiency of the importance sampling scheme, is achieved by the special choice of exponential random graph that is indistinguishable from the Erdős–Rényi graph conditioned to have many triangles. We show how this choice can be made for the conditioned Erdős–Rényi graphs both in the replica symmetric phase and also in parts of the replica breaking phase. Equally interestingly, we also show that the exponential tilt suggested directly by the large deviation principle does not always yield an optimal scheme.

We explicitly compute the exit law of a certain hypoelliptic Brownian motion on a solvable Lie group. The underlying random variable can be seen as a multidimensional exponential functional of Brownian motion. As a consequence, we obtain hidden identities in law between gamma random variables as the probabilistic manifestation of braid relations. The classical beta-gamma algebra identity corresponds to the only braid move in a root system of type $A_2$. The other ones seem new. A key ingredient is a conditional representation theorem. It relates our hypoelliptic Brownian motion conditioned on exiting at a fixed point to a certain deterministic transform of Brownian motion. The identities in law between gamma variables tropicalize to identities between exponential random variables. These are continuous versions of identities between geometric random variables related to changes of parametrizations in Lusztig's canonical basis. Hence, we see that the exit law of our hypoelliptic Brownian motion is the geometric analogue of a simple natural measure on Lusztig's canonical basis.

We propose a new version of Stein's method of exchangeable pairs, which, given a suitable exchangeable pair $(W,W')$ of real-valued random variables, suggests the approximation of the law of $W$ by a suitable absolutely continuous distribution. This distribution is characterized by a first order linear differential Stein operator, whose coefficients $\gamma$ and $\eta$ are motivated by two regression properties satisfied by the pair $(W,W')$. Furthermore, the general theory of Stein's method for such an absolutely continuous distribution is developed and a general characterization result as well as general bounds on the solution to the Stein equation are given. This abstract approach is a certain extension of the theory developed in previous works, which only consider the framework of the density approach, i.e. $\eta\equiv1$. As an illustration of our technique we prove a general plug-in result, which bounds a certain distance of the distribution of a given random variable $W$ to a Beta distribution in terms of a given exchangeable pair $(W,W')$ and provide new bounds on the solution to the Stein equation for the Beta distribution, which complement the existing bounds from previous works. The abstract plug-in result is then applied to derive bounds of order $n^{-1}$ for the distance between the distribution of the relative number of drawn red balls after $n$ drawings in a Pólya urn model and the limiting Beta distribution measured by a certain class of smooth test functions.

We consider in this work a one parameter family of hypoelliptic diffusion processes on the unit tangent bundle $T^1 \mathcal M$ of a Riemannian manifold $(\mathcal M,g)$, collectively called kinetic Brownian motions, that are random perturbations of the geodesic flow, with a parameter $\sigma$ quantifying the size of the noise. Projection on $\mathcal M$ of these processes provides random $C^1$ paths in $\mathcal M$. We show, both qualitively and quantitatively, that the laws of these $\mathcal M$-valued paths provide an interpolation between geodesic and Brownian motions. This qualitative description of kinetic Brownian motion as the parameter $\sigma$ varies is complemented by a thourough study of its long time asymptotic behaviour on rotationally invariant manifolds, when $\sigma$ is fixed, as we are able to give a complete description of its Poisson boundary in geometric terms.

We show that the complex-valued ODE ($n\geq 1$, $a_{n+1} \neq 0$):

$$\dot z = a_{n+1} z^{n+1} + a_n z^n+\cdots+a_0 , $$

which necessarily has trajectories along which the dynamics blows up in finite time, can be stabilized by the addition of an arbitrarily small elliptic, additive Brownian stochastic term. We also show that the stochastic perturbation has a unique invariant probability measure which is heavy-tailed yet is uniformly, exponentially attracting. The methods turn on the construction of Lyapunov functions. The techniques used in the construction are general and can likely be used in other settings where a Lyapunov function is needed. This is a two-part paper. This paper, Part I, focuses on general Lyapunov methods as applied to a special, simplified version of the problem. Part II extends the main results to the general setting.

We study a symmetric diffusion $X$ on $\mathbb{R}^d$ in divergence form in a stationary and ergodic environment, with measurable unbounded and degenerate coefficients. We prove a quenched local central limit theorem for $X$, under some moment conditions on the environment; the key tool is a local parabolic Harnack inequality obtained with Moser iteration technique.

We continue the work started in Part I of this article, showing how the addition of noise can stabilize an otherwise unstable system. The analysis makes use of nearly optimal Lyapunov functions. In this continuation, we remove the main limiting assumption of Part I by an inductive procedure as well as establish a lower bound which shows that our construction is radially sharp. We also prove a version of Peskir's generalized Tanaka formula adapted to patching together Lyapunov functions. This greatly simplifies the analysis used in previous works.

We extend the dimension free Talagrand inequalities for convex distance using an extension of Marton’s weak transport to other metrics than the Hamming distance. We study the dual form of these weak transport inequalities for the euclidian norm and prove that it implies sub-gaussianity and convex Poincaré inequality. We obtain new weak transport inequalities for non products measures extending the results of Samson. Many examples are provided to show that the euclidian norm is an appropriate metric for classical time series. Our approach, based on trajectories coupling, is more efficient to obtain dimension free concentration than existing contractive assumptions. Expressing the concentration properties of the ordinary least square estimator as a conditional weak transport problem, we derive new oracle inequalities with fast rates of convergence in dependent settings.

Consider a long-range percolation model on $\mathbb{Z}^d$ where the probability that an edge $\{x,y\} \in \mathbb{Z}^d \times \mathbb{Z}^d$ is open is proportional to $\|x-y\|_2^{-d \alpha}$ for some $\alpha \gt 0$ and where $d \gt 3 \min\{2,\alpha\}$. We prove that in this case the one-arm exponent equals $\min\{4,\alpha\}/2$. We also prove that the maximal displacement for critical branching random walk scales with the same exponent. This establishes that both models undergo a phase transition in the parameter $\alpha$ when $\alpha =4$.

We study competition of two spreading colors starting from single sources on the configuration model with i.i.d. degrees following a power-law distribution with exponent $\tau\in (2,3)$. In this model two colors spread with a fixed but not necessarily equal speed on the unweighted random graph. We show that if the speeds are not equal, then the faster color paints almost all vertices, while the slower color can paint only a random subpolynomial fraction of the vertices. We investigate the case when the speeds are equal and typical distances in a follow-up paper.

We consider Lipschitz percolation in $d+1$ dimensions above planes tilted by an angle $\gamma$ along one or several coordinate axes. In particular, we are interested in the asymptotics of the critical probability as $d \to \infty$ as well as $\gamma \uparrow \pi/4.$ Our principal results show that the convergence of the critical probability to 1 is polynomial as $d\to \infty$ and $\gamma \uparrow \pi/4.$ In addition, we identify the correct order of this polynomial convergence and in $d=1$ we also obtain the correct prefactor.

We obtain sharp asymptotic estimates for hitting probabilities of a critical branching Brownian motion in one dimension with killing at 0. We also obtain sharp asymptotic formulas for the tail probabilities of the number of particles killed at 0. In the special case of double-or-nothing branching, we give exact formulas for both the hitting probabilities, in terms of elliptic functions, and the distribution of the number of killed particles.

The notion of stability can be generalised to point processes by defining the scaling operation in a randomised way: scaling a configuration by $t$ corresponds to letting such a configuration evolve according to a Markov branching particle system for $-\log t$ time. We prove that these are the only stochastic operations satisfying basic associativity and distributivity properties and we thus introduce the notion of branching-stable point processes. For scaling operations corresponding to particles that branch but do not diffuse, we characterise stable distributions as thinning stable point processes with multiplicities given by the quasi stationary (or Yaglom) distribution of the branching process under consideration. Finally we extend branching-stability to continuous random variables with the help of continuous branching (CB) processes, and we show that, at least in some frameworks, branching-stable integer random variables are exactly Cox (doubly stochastic Poisson) random variables driven by corresponding CB-stable continuous random variables.

We study random points on the real line generated by the eigenvalues in unitary invariant random matrix ensembles or by more general repulsive particle systems. As the number of points tends to infinity, we prove convergence of the empirical distribution of nearest neighbor spacings. We extend existing results for the spacing distribution in two ways. On the one hand, we believe the empirical distribution to be of more practical relevance than the so far considered expected distribution. On the other hand, we use the unfolding, a non-linear rescaling, which transforms the ensemble such that the density of particles is asymptotically constant. This allows to consider all empirical spacings, where previous results were restricted to a tiny fraction of the particles. Moreover, we prove bounds on the rates of convergence. The main ingredient for the proof, a strong bulk universality result for correlation functions in the unfolded setting including optimal rates, should be of independent interest.

The paper proves several limit theorems for linear eigenvalue statistics of overlapping Wigner and sample covariance matrices. It is shown that the covariance of the limiting multivariate Gaussian distribution is diagonalized by choosing the Chebyshev polynomials of the first kind as the basis for the test function space. The covariance is explicitly computed and it is shown that for the Chebyshev polynomials of sufficiently high degree the covariance of linear statistics depends only on the first two moments of matrix entries.

We study the hitting times of Markov processes to target set $G$, starting from a reference configuration $x_0$ or its basin of attraction. The configuration $x_0$ can correspond to the bottom of a (meta)stable well, while the target $G$ could be either a set of saddle (exit) points of the well, or a set of further (meta)stable configurations. Three types of results are reported: (1) A general theory is developed, based on the path-wise approach to metastability, which has three important attributes. First, it is general in that it does not assume reversibility of the process, does not focus only on hitting times to rare events and does not assume a particular starting measure. Second, it relies only on the natural hypothesis that the mean hitting time to $G$ is asymptotically longer than the mean recurrence time to $x_0$ or $G$. Third, despite its mathematical simplicity, the approach yields precise and explicit bounds on the corrections to exponentiality. (2) We compare and relate different metastability conditions proposed in the literature so to eliminate potential sources of confusion. This is specially relevant for evolutions of infinite-volume systems, whose treatment depends on whether and how relevant parameters (temperature, fields) are adjusted. (3) We introduce the notion of early asymptotic exponential behavior to control time scales asymptotically smaller than the mean-time scale. This control is particularly relevant for systems with unbounded state space where nucleations leading to exit from metastability can happen anywhere in the volume. We provide natural sufficient conditions on recurrence times for this early exponentiality to hold and show that it leads to estimations of probability density functions.

We combine Malliavin calculus with Stein's method to derive bounds for the Variance-Gamma approximation of functionals of isonormal Gaussian processes, in particular of random variables living inside a fixed Wiener chaos induced by such a process. The bounds are presented in terms of Malliavin operators and norms of contractions. We show that a sequence of distributions of random variables in the second Wiener chaos converges to a Variance-Gamma distribution if and only if their moments of order two to six converge to that of a Variance-Gamma distributed random variable (six moment theorem). Moreover, simplified versions for Laplace or symmetrized Gamma distributions are presented. Also multivariate extensions and a universality result for homogeneous sums are considered.

For a complete graph of size $n$, assign each edge an i.i.d. exponential variable with mean $n$. For $\lambda \gt 0$, consider the length of the longest path whose average weight is at most $\lambda$. It was shown by Aldous (1998) that the length is of order $\log n$ for $\lambda \lt 1/\mathrm{e}$ and of order $n$ for $\lambda \gt 1/\mathrm{e}$. In this paper, we study the near-supercritical regime where $\lambda = \mathrm{e}^{-1} +\eta$ with $\eta \gt 0$ a small fixed number. We show that there exist two absolute constants $C_1, C_2 \gt 0$ such that with high probability the length is in between $n \mathrm{e}^{-C_1/\sqrt{\eta}}$ and $n \mathrm{e}^{-C_2/\sqrt{\eta}}$. Our result corrects a non-rigorous prediction of Aldous (2005).

The continuum random cluster model is defined as a Gibbs modification of the stationary Boolean model in $\mathbb{R}^d$ with intensity $z \gt 0$ and the law of radii $Q$. The formal unormalized density is given by $q^{N_{cc}}$ where $q \gt 0$ is a fixed parameter and $N_{cc}$ the number of connected components in the random germ-grain structure. In this paper we prove the existence of the model in the infinite volume regime for a large class of parameters including the case $q \lt 1$ or distributions $Q$ without compact support. In the extreme setting of non integrable radii (i.e. $\int R^d Q(dR)=\infty$) and $q$ is an integer larger than 1, we prove that for $z$ small enough the continuum random cluster model is not unique; two different probability measures solve the DLR equations. We conjecture that the uniqueness is recovered for $z$ large enough which would provide a phase transition result. Heuristic arguments are given. Our main tools are the compactness of level sets of the specific entropy, a fine study of the quasi locality of the Gibbs kernels and a Fortuin-Kasteleyn representation via Widom-Rowlinson models with random radii.

The order $O_n(\sigma)$ of a permutation $\sigma$ of $n$ objects is the smallest integer $k \geq 1$ such that the $k$-th iterate of $\sigma$ gives the identity. A remarkable result about the order of a uniformly chosen permutation is due to Erdös and Turán who proved in 1965 that $\log O_n$ satisfies a central limit theorem. We extend this result to the so-called generalized Ewens measure in a previous paper. In this paper, we establish a local limit theorem as well as, under some extra moment condition, a precise large deviations estimate. These properties are new even for the uniform measure. Furthermore, we provide precise large deviations estimates for random permutations with polynomial cycle weights.

We solve the Skorokhod embedding problem for a class of Gaussian processes including Brownian motion with non-linear drift. Our approach relies on solving an associated strongly coupled system of Forward Backward Stochastic Differential Equation (FBSDE), and investigating the regularity of the obtained solution. For this purpose we extend the existence, uniqueness and regularity theory of so called decoupling fields for Markovian FBSDE to a setting in which the coefficients are only locally Lipschitz continuous.

Explicit rate of convergence in variance (or more general entropies) is obtained for a class of Piecewise Deterministic Markov Processes such as the TCP process, relying on functional inequalities. A method to establish Poincaré (and more generally Beckner) inequalities with respect to a diffusion-type energy for the invariant law of such hybrid processes is developed.

We study a large class of Bernoulli percolation models on random lattices of the half-plane, obtained as local limits of uniform planar triangulations or quadrangulations. We first compute the exact value of the site percolation threshold in the quadrangular case using the so-called peeling techniques. Then, we generalize a result of Angel about the scaling limit of crossing probabilities, that are a natural analogue to Cardy’s formula in (non-random) plane lattices. Our main result is that those probabilities are universal, in the sense that they do not depend on the percolation model neither on the degree of the faces of the map.