In this note we first consider local times of random walks killed at leaving positive half-axis. We prove that the distribution of the properly rescaled local time at point $N$ conditioned on being positive converges towards an exponential distribution. The proof is based on known results for conditioned random walks, which allow to determine the asymptotic behaviour of moments of local times. Using this information we also show that the field of local times of a reflected random walk converges in the sense of finite dimensional distributions. This is in the spirit of the seminal result by Knight [10] who has shown that for the symmetric simple random walk local times converge weakly towards a squared Bessel process. Our result can be seen as an extension of the second Ray-Knight theorem to all asymptotically stable random walks.

We consider the serve-the-longest-queue discipline for a multiclass queue with buffers of equal size, operating under (i) the conventional and (ii) the Halfin-Whitt heavy traffic regimes, and show that while the queue length process’ scaling limits are fully determined by the first and second order data in case (i), they depend on finer properties in case (ii). The proof of the latter relies on the construction of a deterministic arrival pattern.

We consider a non-Markovian optimal stopping problem on finite horizon. We prove that the value process can be represented by means of a backward stochastic differential equation (BSDE), defined on an enlarged probability space, containing a stochastic integral having a one-jump point process as integrator and an (unknown) process with a sign constraint as integrand. This provides an alternative representation with respect to the classical one given by a reflected BSDE. The connection between the two BSDEs is also clarified. Finally, we prove that the value of the optimal stopping problem is the same as the value of an auxiliary optimization problem where the intensity of the point process is controlled.

We prove the occurrence of a phase transition accompanied by the emergence of cycles of diverging lengths in the random interchange process on the hypercube.

We consider time-dependent random walks among time-dependent conductances. For discrete time random walks, we show that, unlike the time-independent case, two-sided Gaussian heat kernel estimates are not stable under perturbations. This is proved by giving an example of a ballistic and transient time-dependent random walk on $\mathbb{Z}$ among uniformly elliptic time-dependent conductances. For continuous time random walks, we show the instability when the holding times are i.i.d. $\exp(1)$, and in contrast, we prove the stability when the holding times change by sites in such a way that the base measure is a uniform measure.

It is well known that if $({X}_{1},...,{X}_{n})$ are i.i.d. r.v.'s taken from either the exponential distribution or the geometric one, then the distribution of $\min({X}_{1},...,{X}_{n})$ is, with a change of parameter, is also exponential or geometric, respectively. In this note we prove the following result. Let $F$ be a natural exponential family (NEF) on $\mathbb{R}$ generated by an arbitrary positive Radon measure $\mu$ (not necessarily confined to the Lebesgue or counting measures on $\mathbb{R}$). Consider $n$ i.i.d. r.v.'s $({X}_{1},...,{X}_{n})$, $n \in 2$, taken from $F$ and let $Y =\min({X}_{1},...,{X}_{n})$. We prove that the family $G$ of distributions induced by $Y$ constitutes an NEF if and only if, up to an affine transformation, $F$ is the family of either the exponential distributions or the geometric distributions. The proof of such a result is rather intricate and probabilistic in nature.

We recover the pathwise Itô solution (the solution to a rough differential equation driven by the Itô signature) by concatenating averaged Stratonovich solutions on small intervals and by letting the mesh of the partition in the approximations tend to zero. More specifically, on a fixed small interval, we consider two Stratonovich solutions: one is driven by the original process and the other is driven by the original process plus a selected independent noise. Then by taking the expectation with respect to the selected noise, we can recover the increment of the bracket process and so recover the leading order approximation of the Itô solution up to a small error. By concatenating averaged increments and by letting the mesh tend to zero, the error tends to zero and we recover the Itô solution.

For a finite undirected graph $G = (V,E)$, let $p_{u,v}(t)$ denote the probability that a continuous-time random walk starting at vertex $u$ is in $v$ at time $t$. In this note we give an example of a Cayley graph $G$ and two vertices $u,v \in G$ for which the function \[ r_{u,v}(t) = \frac{p_{u,v}(t)} {p_{u,u}(t)} \qquad t \geq 0 \] is not monotonically non-decreasing. This answers a question asked by Peres in 2013.

In the standard nearest-neighbor coarsening model with state space $\{-1,+1\}^{\mathbb{Z} ^2}$ and initial state chosen from symmetric product measure, it is known (see [2]) that almost surely, every vertex flips infinitely often. In this paper, we study the modified model in which a single vertex is frozen to $+1$ for all time, and show that every other site still flips infinitely often. The proof combines stochastic domination (attractivity) and influence propagation arguments.

The variational principle for Gibbs point processes with general finite range interaction is proved. Namely, the Gibbs point processes are identified as the minimizers of the free excess energy equals to the sum of the specific entropy and the mean energy. The interaction is very general and includes superstable pairwise potential, finite or infinite multibody potential, geometrical interaction, hardcore interaction. The only restrictive assumption involves the finite range property.

We provide a new proof of a Kramers’ type law for self-stabilizing diffusion. These diffusions correspond to the hydrodynamical limit of a mean-field system of particles and may be seen as the probabilistic interpretation of the granular media equation. We use the same hypotheses as the ones used in the work “Large deviations and a Kramers’ type law for self-stabilizing diffusions” by Herrmann, Imkeller and Peithmann in which the authors obtain a first proof of the statement.

We derive the strongest individual fitness distribution on a variation for a species survival model proposed by Guiol, Machado and Schinazi [5]. We point out to the fact that this distribution relies on the Gauss hypergeometric function and when $p=\frac{1} {2}$ on the hypergeometric function type I distribution.

We make a few elementary observations that relate directly the items mentioned in the title. In particular, we note that when one superimposes the random current model related to the Ising model with an independent Bernoulli percolation model with well-chosen weights, one obtains exactly the FK-percolation (or random cluster model) associated with the Ising model, and we point out that this relation can be interpreted via loop-soups, combining the description of the sign of a Gaussian free field on a discrete graph knowing its square (and the relation of this question with the FK-Ising model) with the loop-soup interpretation of the random current model.

It was proved by Hoeffding in 1963 that a real random variable $X$ confined to $[a,b]$ satisfies $\mathbb{E} \, e^{X-\operatorname{\mathbb {E}} X} \le e^{(b-a)^2/8}$. We generalise this to complex random variables.

We prove the trichotomy between transience to the right, transience to the left and recurrence of one-dimensional nearest-neighbour random walks in dynamic random environments under fairly general assumptions, namely: stationarity under space-time translations, ergodicity under spatial translations, and a mild ellipticity condition. In particular, the result applies to general uniformly elliptic models and also to a large class of non-uniformly elliptic cases that are i.i.d. in space and Markovian in time. An immediate consequence is the recurrence of models that are symmetric with respect to reflection through the origin.

We propose a discrete analogue for the boundary local time of reflected diffusions in bounded Lipschitz domains. This discrete analogue, called the discrete local time, can be effectively simulated in practice and is obtained pathwise from random walks on lattices. We establish weak convergence of the joint law of the discrete local time and the associated random walks as the lattice size decreases to zero. A cornerstone of the proof is the local central limit theorem for reflected diffusions developed in [7]. Applications of the join convergence result to PDE problems are illustrated.

The pointwise asymptotic properties of the Parzen-Rosenblatt kernel estimator $\widehat{f} _n$ of a probability density function $f$ on $\mathbb{R} ^d$ have received great attention, and so have its integrated or uniform errors. It has been pointed out in a couple of recent works that the weak convergence of its centered and rescaled versions in a weighted Lebesgue $L^p$ space, $1\leq p<\infty $, considered to be a difficult problem, is in fact essentially uninteresting in the sense that the only possible Borel measurable weak limit is 0 under very mild conditions. This paper examines the weak convergence of such processes in the uniform topology. Specifically, we show that if $f_n(x)=\mathbb{E} (\widehat{f} _n(x))$ and $(r_n)$ is any nonrandom sequence of positive real numbers such that $r_n/\sqrt{n} \to 0$ then, with probability 1, the sample paths of any tight Borel measurable weak limit in an $\ell ^{\infty }$ space on $\mathbb{R} ^d$ of the process $r_n(\widehat{f} _n-f_n)$ must be almost everywhere zero. The particular case when the estimator $\widehat{f} _n$ has continuous sample paths is then considered and simple conditions making it possible to examine the actual existence of a weak limit in this framework are provided.

In this paper we consider fractional higher-order stochastic differential equations of the form \[ \left ( \mu + c_\alpha \frac{d^\alpha } {dt^\alpha } \right )^\beta X(t) = \mathcal{E} (t) , \quad \mu >0,\; \beta >0,\; \alpha \in (0,1) \cup \mathbb{N} \] where $\mathcal{E} (t)$ is a Gaussian white noise. We obtain explicitly the covariance functions and the spectral densities of the stochastic processes satisfying the above equations.

In this paper we investigate the behavior of the bridges of a Markov counting process in several directions. We first characterize convexity(concavity) in time of the mean value in terms of lower (upper) bounds on the so called reciprocal characteristics. This result gives a natural criterion to determine whether bridges are “lazy” or “hurried”. Under the hypothesis of global bounds on the reciprocal characteristics we prove sharp estimates for the marginal distributions and a comparison theorem for the jump times. When the height of the bridge tends to infinity we show the convergence to a deterministic curve, after a proper rescaling.

The space-fractional Poisson process is a time-changed homogeneous Poisson process where the time change is an independent stable subordinator. In this paper, a further generalization is discussed that preserves the Lévy property. We introduce a generalized process by suitably time-changing a superposition of weighted space-fractional Poisson processes. This generalized process can be related to a specific subordinator for which it is possible to explicitly write the characterizing Lévy measure. Connections are highlighted to Prabhakar derivatives, specific convolution-type integral operators. Finally, we study the effect of introducing Prabhakar derivatives also in time.

We study the propagation of high peaks (intermittency fronts) of the solution to a stochastic heat equation driven by multiplicative centered Gaussian noise in $\mathbb{R} ^d$. The noise is assumed to have a general homogeneous covariance in both time and space, and the solution is interpreted in the senses of the Wick product. We give some estimates for the upper and lower bounds of the propagation speed, based on a moment formula of the solution. When the space covariance is given by a Riesz kernel, we give more precise bounds for the propagation speed.

A one-dimensional, continuous, regular, and strong Markov process $X$ with state space $E$ hits any point $z \in E$ fast with positive probability. To wit, if ${\boldsymbol{\tau } }_z = \inf \{t \geq 0:X_{t} = z\}$, then $\textsf{P} _\xi ({\boldsymbol{\tau } }_z<\varepsilon )>0$ for all $\xi \in E$ and $\varepsilon >0$.

We establish a weighted maximal $L^1$-inequality for differentially subordinate martingales taking values in $\mathbb{R} ^\nu $, $\nu \geq 1$, under the assumption that the weight satisfies Muckenhoupt’s condition $A_1$. An optimal dependence of the constant on the $A_1$ characteristics is identified.

A conditioned hypoelliptic process on a compact manifold, satisfying the strong Hörmander’s condition, is a hypoelliptic bridge. If the Markov generator satisfies the two step strong Hörmander condition, the drift of the conditioned hypoelliptic bridge is integrable on $[0,1]$ and the hypoelliptic bridge is a continuous semi-martingale.

Comparison and converse comparison theorems are important parts of the research on backward stochastic differential equations. In this paper, we obtain comparison results for one dimensional backward stochastic differential equations with Markov chain noise, adapting previous results under simplified hypotheses. We introduce a type of nonlinear expectation, the $f$-expectation, which is an interpretation of the solution to a BSDE, and use it to establish a converse comparison theorem for the same type of equations as those in the comparison results.

Consider the subgraph of the discrete $d$-dimensional torus of size length $N$, $d\geq 3$, induced by the range of the simple random walk on the torus run until the time $uN^d$. We prove that for all $d\geq 3$ and $u>0$, the mixing time for the random walk on this subgraph is of order $N^2$ with probability at least $1 - Ce^{-(\log N)^2}$.

We study the connectivity of random subgraphs of the $d$-dimensional Hamming graph $H(d, n)$, which is the Cartesian product of $d$ complete graphs on $n$ vertices. We sample the random subgraph with an i.i.d. Bernoulli bond percolation on $H(d,n)$ with parameter $p$. We identify the window of the transition: when $ np- \log n \to - \infty $ the probability that the graph is connected tends to $0$, while when $ np- \log n \to + \infty $ it converges to $1$. We also investigate the connectivity probability inside the critical window, namely when $ np- \log n \to t \in \mathbb{R} $. We find that the threshold does not depend on $d$, unlike the phase transition of the giant connected component of the Hamming graph (see [1]). Within the critical window, the connectivity probability does depend on $d$. We determine how.

We study critical percolation on a regular planar lattice. Let $E_G(n)$ be the expected number of open clusters intersecting or hitting the line segment $[0,n]$. (For the subscript $G$ we either take $\mathbb{H} $, when we restrict to the upper halfplane, or $\mathbb{C} $, when we consider the full lattice).

Cardy [2] (see also Yu, Saleur and Haas [11]) derived heuristically that $E_{\mathbb{H} }(n) = An + \frac{\sqrt {3}} {4\pi }\log (n) + o(\log (n))$, where $A$ is some constant. Recently Kovács, Iglói and Cardy derived in [5] heuristically (as a special case of a more general formula) that a similar result holds for $E_{\mathbb{C} }(n)$ with the constant $\frac{\sqrt {3}} {4\pi }$ replaced by $\frac{5\sqrt {3}} {32\pi }$.

In this paper we give, for site percolation on the triangular lattice, a rigorous proof for the formula of $E_{\mathbb{H} }(n)$ above, and a rigorous upper bound for the prefactor of the logarithm in the formula of $E_{\mathbb{C} }(n)$.

Let $X$ be a symmetric, pure jump, unimodal Lévy process in $\mathbb{R} $ with an infinite Lévy measure. We prove that for any fixed $t > 0$ the survival probability $P^x(\tau _{(-a,a)} > t)$ is nondecreasing on $(-a,0]$, nonincreasing on $[0,a)$ and concave on $(-a/2,a/2)$, where $a > 0$ and $\tau _{(-a,a)}$ is the first exit time of the process $X$ from $(-a,a)$. We also show a similar statement for sets $(-a,a) \times F \subset \mathbb{R} ^d$.

We consider the random walk loop soup on the discrete half-plane $\mathbb{Z} \times \mathbb{N} ^{\ast }$ and study the percolation problem, i.e. the existence of an infinite cluster of loops. We show that the critical value of the intensity is equal to $\frac{1} {2}$. The absence of percolation at intensity $\frac{1} {2}$ was shown in a previous work. We also show that in the supercritical regime, one can keep only the loops up to some large enough upper bound on the diameter and still have percolation.

We are interested in the increment stationarity property of $L^2$-indexed stochastic processes, which is a fairly general concern since many random fields can be interpreted as the restriction of a more generally defined $L^2$-indexed process. We first give a spectral representation theorem in the sense of Ito [9], and see potential applications on random fields, in particular on the $L^2$-indexed extension of the fractional Brownian motion. Then we prove that this latter process is characterized by its increment stationarity and self-similarity properties, as in the one-dimensional case.

This erratum corrects a piece in the proof of the lower bound for an Erdős-Rényi law of large numbers in the nonconventional setup obtained in [1].

Skorokhod’s M1 topology is defined for càdlàg paths taking values in the space of tempered distributions (more generally, in the dual of a countably Hilbertian nuclear space). Compactness and tightness characterisations are derived which allow us to study a collection of stochastic processes through their projections on the familiar space of real-valued càdlàg processes. It is shown how this topological space can be used in analysing the convergence of empirical process approximations to distribution-valued evolution equations with Dirichlet boundary conditions.

In this note, we compute the Mellin transform of a Poissonian exponential functional, the underlying process being a simple continuous time random walk. It shows that the Poissonian functional can be expressed in term of the inverse of a $q$-gamma random variable.

The result interpolates between two known results. When the random walk has only positive increments, we retrieve a theorem due to Bertoin, Biane and Yor. In the Brownian limit ($q \rightarrow 1^-$), one recovers Dufresne’s identity involving an inverse gamma random variable. Hence, one can see it as a $q$-deformed Dufresne identity.

The nature of the alignment with gaps corresponding to a longest common subsequence (LCS) of two independent iid random sequences drawn from a finite alphabet is investigated. It is shown that such an optimal alignment typically matches pieces of similar short-length. This is of importance in understanding the structure of optimal alignments of two sequences. Moreover, it is also shown that any property, common to two subsequences, typically holds in most parts of the optimal alignment whenever this same property holds, with high probability, for strings of similar short-length. Our results should, in particular, prove useful for simulations since they imply that the re-scaled two dimensional representation of a LCS gets uniformly close to the diagonal as the length of the sequences grows without bound.

In this article we propose a model for stochastic delay differential equation with jumps (SDDEJ) in a differentiable manifold $M$ endowed with a connection $\nabla $. In our model, the continuous part is driven by vector fields with a fixed delay and the jumps are assumed to come from a distinct source of (càdlàg) noise, without delay. The jumps occur along adopted differentiable curves with some dynamical relevance (with fictitious time) which allow to take parallel transport along them. In the last section, using a geometrical approach, we show that the horizontal lift of the solution of an SDDEJ is again a solution of an SDDEJ in the linear frame bundle $BM$ with respect to a horizontal connection $\nabla ^H$ in $BM$.

We present two examples of a large deviations principle where the rate function is not strictly convex. This is motivated by a model used in mathematical finance (the Heston model), and adds a new item to the zoology of non strictly convex large deviations.

Consider the following forest-fire model on the upper half-plane of the triangular lattice: Each site can be “vacant” or “occupied by a tree”. At time $0$ all sites are vacant. Then the process is governed by the following random dynamics: Trees grow at rate $1$, independently for all sites. If an occupied cluster reaches the boundary of the half-plane or if it is about to become infinite, the cluster is instantaneously destroyed, i.e. all of its sites turn vacant.

Let $t_c = \log 2$ denote the critical time after which an infinite cluster first appears in the corresponding pure growth process, where there is only the growth of trees but no destruction mechanism. Choose an arbitrary infinite cone in the half-plane whose apex lies on the boundary of the half-plane and whose boundary lines are non-horizontal. We prove that at time $t_c$ almost surely only finitely many sites inside the cone have been affected by destruction in the forest-fire process.

We consider the empirical eigenvalue distribution of random real symmetric matrices with stochastically independent skew-diagonals and study its limit if the matrix size tends to infinity. We allow correlations between entries on the same skew-diagonal and we distinguish between two types of such correlations, a rather weak and a rather strong one. For weak correlations the limiting distribution is Wigner’s semi-circle distribution; for strong correlations it is the free convolution of the semi-circle distribution and the limiting distribution for random Hankel matrices.

Let $U=(U_k)_{k\in{\mathbb Z} }$ be a centered Gaussian stationary sequence satisfying some minor regularity condition. We study the asymptotic behavior of its weighted $\ell _2$-norm small deviation probabilities. It is shown that \[ \ln \mathbb{P} \left ( \sum _{k\in \mathbb{Z} } d_k^2 U_k^2 \le \varepsilon ^2\right ) \sim - M \varepsilon ^{-\frac{2} {2p-1}}, \qquad \textrm{ as } \varepsilon \to 0, \] whenever \[ d_k\sim d_{\pm } |k|^{-p}\quad \textrm{for some } p>\frac{1} {2} \, , \quad k\to \pm \infty , \] using the arguments based on the spectral theory of pseudo-differential operators by M. Birman and M. Solomyak. The constant $M$ reflects the dependence structure of $U$ in a non-trivial way, and marks the difference with the well-studied case of the i.i.d. sequences.

We generalize the exactly solvable corner growth models by choosing the rate of the exponential distribution $a_i+b_j$ and the parameter of the geometric distribution $a_i b_j$ at site $(i, j)$, where $(a_i)_{i \ge 1}$ and $(b_j)_{j \ge 1}$ are jointly ergodic random sequences. We identify the shape function in terms of a simple variational problem, which can be solved explicitly in some special cases.

Let $\tau (x)$ be the first time that the reflected process $Y$ of a Lévy process $X$ crosses $x>0$. The main aim of this paper is to investigate the joint asymptotic distribution of $Y(t)=X(t) - \inf _{0\leq s\leq t}X(s)$ and the path functionals $Z(x)=Y(\tau (x))-x$ and $m(t)=\sup _{0\leq s\leq t}Y(s) - y^*(t)$, for a certain non-linear curve $y^*(t)$. We restrict to Lévy processes $X$ satisfying Cramér’s condition, a non-lattice condition and the moment conditions that $E[|X(1)|]$ and $E[\exp (\gamma X(1))|X(1)|]$ are finite (where $\gamma $ denotes the Cramér coefficient). We prove that $Y(t)$ and $Z(x)$ are asymptotically independent as $\min \{t,x\}\to \infty $ and characterise the law of the limit $(Y_\infty ,Z_\infty )$. Moreover, if $y^*(t) = \gamma ^{-1}\log (t)$ and $\min \{t,x\}\to \infty $ in such a way that $t\exp \{-\gamma x\}\to 0$, then we show that $Y(t)$, $Z(x)$ and $m(t)$ are asymptotically independent and derive the explicit form of the joint weak limit $(Y_\infty , Z_\infty , m_\infty )$. The proof is based on excursion theory, Theorem 1 in [7] and our characterisation of the law $(Y_\infty , Z_\infty )$.

We consider a $q$-deformed version of the uniform Gibbs measure on dimers on the periodized hexagonal lattice (equivalently, on interlacing particle configurations, if vertical dimers are seen as particles) and show that it is invariant under a certain irreversible $q$-Whittaker dynamic. Thereby we provide a new non-trivial example of driven interacting two-dimensional particle system, or of $(2+1)$-dimensional stochastic growth model, with explicit stationary measure. We emphasize that this measure is far from being a product Bernoulli measure. These Gibbs measures and dynamics both arose earlier in the theory of Macdonald processes [7]. The $q=0$ degeneration of the Gibbs measures reduce to the usual uniform dimer measures with given tilt [12], the degeneration of the dynamics originate in the study of Schur processes [5, 6] and the degeneration of the results contained herein were recently treated in [19].

We consider a particular instance of the truncated realizability problem on the $d-$dimensional lattice. Namely, given two functions $\rho _1({\bf i})$ and $\rho _2({\bf i},{\bf j})$ non-negative and symmetric on $\mathbb{Z} ^d$, we ask whether they are the first two correlation functions of a translation invariant point process. We provide an explicit construction of such a realizing process for any $d\geq 2$ when the radial distribution has a specific form. We also derive from this construction a lower bound for the maximal realizable density and compare it with the already known lower bounds.

We show that there is a sharp threshold in dimension one for the transport cost between the Lebesgue measure $\lambda $ and an invariant random measure $\mu $ of unit intensity to be finite. We show that for any such random measure the $L^1$ cost is infinite provided that the first central moments $\mathbb{E} [|n-\mu ([0,n))|]$ diverge. Furthermore, we establish simple and sharp criteria, based on the variance of $\mu ([0,n)]$, for the $L^p$ cost to be finite for $0<p<1$.

Begin continuous time random walks from every vertex of a graph and have particles coalesce when they collide. We use a duality relation with the voter model to prove the process is site recurrent on bounded degree graphs, and for Galton-Watson trees whose offspring distribution has exponential tail. We prove bounds on the occupation probability of a site, as well as a general 0-1 law. Similar conclusions hold for a coalescing process on trees where particles do not backtrack.

We prove a version of the multidimensional Fourth Moment Theorem for chaotic random vectors, in the general context of diffusion Markov generators. In addition to the usual componentwise convergence and unlike the infinite-dimensional Ornstein-Uhlenbeck generator case, another moment-type condition is required to imply joint convergence of of a given sequence of vectors.

The asymptotic law of the truncated $S\times S$ random submatrix of a Haar random matrix in $\mathrm{GL}_N(\mathbb{Z}_m)$ as $N$ goes to infinity is obtained. The same result is also obtained when $\mathbb{Z}_m$ is replaced by any commutative compact local ring whose maximal ideal is topologically closed.

We investigate the transition density of a hyperbolic Bessel process for integer dimensions and show a link between transition densities of a hyperbolic Brownian motion and a Bessel process in the same dimension. Using the so-called Millson’s formula for the densities of hyperbolic Brownian motion we also show a link between transition density of $n$-dimensional hyperbolic Bessel process and $2$-dimensional (if $n$ is even) or $3$-dimensional (if $n$ is odd) hyperbolic Brownian motion. This helps us to get explicit formulas for the Bessel process transition density.

Consider the perpetuity equation $X \stackrel{\mathcal {D}} {=} A X + B$, where $(A,B)$ and $X$ on the right-hand side are independent. The Kesten–Grincevičius–Goldie theorem states that if $\mathbf{E} A^\kappa = 1$, $\mathbf{E} A^\kappa \log _+ A < \infty $, and $\mathbf{E} |B|^\kappa < \infty $, then $\mathbf{P} \{ X > x \} \sim c x^{-\kappa }$. Assume that $\mathbf{E} |B|^\nu < \infty $ for some $\nu > \kappa $, and consider two cases (i) $\mathbf{E} A^\kappa = 1$, $\mathbf{E} A^\kappa \log _+ A = \infty $; (ii) $\mathbf{E} A^\kappa < 1$, $\mathbf{E} A^t = \infty $ for all $t > \kappa $. We show that under appropriate additional assumptions on $A$ the asymptotic $\mathbf{P} \{ X > x \} \sim c x^{-\kappa } \ell (x) $ holds, where $\ell $ is a nonconstant slowly varying function. We use Goldie’s renewal theoretic approach.

In a recent paper by Gorin and Shkolnikov (2016), they have found, as a corollary to their result relevant to random matrix theory, that the area below a normalized Brownian excursion minus one half of the integral of the square of its total local time, is identical in law with a centered Gaussian random variable with variance $1/12$. In this paper, we give a pathwise interpretation to their identity; Jeulin’s identity connecting normalized Brownian excursion and its local time plays an essential role in the exposition.

This paper proves almost-sure convergence for the self attracting diffusion on the unit sphere \[ dX_t=\nu \circ dW_{t}(X_t)-a\int _{0}^{t}\nabla _{\mathbb{S} ^n}V_{X_s}(X_t) dsdt,\qquad X_0=x\in \mathbb{S} ^n, \] where $\nu >0$, $a < 0$, $V_y(x)=\langle x,y\rangle $ is the usual scalar product on $\mathbb{R} ^{n+1}$, $\circ $ stands for the Stratonovich differential and $(W_{t}(.))_{t\geqslant 0}$ is a Brownian vector field on $\mathbb{S} ^n$. From this we deduce the almost-sure convergence of the real-valued self attracting diffusion \[ d\vartheta _{t}=\nu dW_{t}+a\int _{0}^{t}\sin (c(\vartheta _{t}-\vartheta _{s}))dsdt, \] where $(W_t)_{t\geqslant 0}$ is a real Brownian motion and $c>0$.

We give a construction of a tree in which the contact process with any positive infection rate survives but, if a certain privileged edge $e^*$ is removed, one obtains two subtrees in which the contact process with infection rate smaller than $1/4$ dies out.

In this note, we prove a multidimensional counterpart of the central limit theorem on the free Poisson chaos recently proved by Bourguin and Peccati (2014). A noteworthy property of convergence toward the semicircular distribution on the free Poisson chaos is obtained as part of the limit theorem: component-wise convergence of sequences of vectors of multiple integrals with respect to a free Poisson random measure toward the semicircular distribution implies joint convergence. This result complements similar findings for the Wiener chaos by Peccati and Tudor (2005), the classical Poisson chaos by Peccati and Zheng (2010) and the Wigner chaos by Nourdin, Peccati and Speicher (2013).

We establish uniform bounds on the low-order derivatives of Stein equation solutions for a broad class of multivariate, strongly log-concave target distributions. These “Stein factor” bounds deliver control over Wasserstein and related smooth function distances and are well-suited to analyzing the computable Stein discrepancy measures of Gorham and Mackey. Our arguments of proof are probabilistic and feature the synchronous coupling of multiple overdamped Langevin diffusions.

We consider the abelian sandpile model and the uniform spanning unicycle on random planar maps. We show that the sandpile density converges to 5/2 as the maps get large. For the spanning unicycle, we show that the length and area of the cycle converges to the exit location and exit time of a simple random walk in the first quadrant. The calculations use the “hamburger-cheeseburger” construction of Fortuin–Kasteleyn random cluster configurations on random planar maps.

We compute an asymptotic expansion with precision $1/n$ of the moments of the expected empirical spectral measure of Wigner matrices of size $n$ with independent centered entries. We interpret this expansion as the moments of the addition of the semi-circle law and $1/n$ times an explicit signed measured with null total mass. This signed measure depends only on the second and fourth moments of the entries.

Using graphical methods based on a ‘lookdown’ and pruned version of the ancestral selection graph, we obtain a representation of the type distribution of the ancestor in a two-type Wright-Fisher population with mutation and selection, conditional on the overall type frequency in the old population. This extends results from [17] to the case of heavy-tailed offspring, directed by a reproduction measure $\Lambda$. The representation is in terms of the equilibrium tail probabilities of the line-counting process $L$ of the graph. We identify a strong pathwise Siegmund dual of $L$, and characterise the equilibrium tail probabilities of $L$ in terms of hitting probabilities of the dual process.

We prove general results about separation and weak$^{\#}$-convergence of boundedly finite measures on separable metric spaces and Souslin spaces. More precisely, we consider an algebra of bounded real-valued, or more generally a $*$-algebra $\mathcal{F} $ of bounded complex-valued functions and give conditions for it to be separating or weak$^\#$-convergence determining for those boundedly finite measures that integrate all functions in $\mathcal{F} $. For separation, it is sufficient if $\mathcal{F} $ separates points, vanishes nowhere, and either consists of only countably many measurable functions, or of arbitrarily many continuous functions. For convergence determining, it is sufficient if $\mathcal{F} $ induces the topology of the underlying space, and every bounded set $A$ admits a function in $\mathcal{F} $ with values bounded away from zero on $A$.

We consider the smoothed multiplicative noise stochastic heat equation \[\mathrm{d} u_{\varepsilon ,t}= \frac 12 \Delta u_{\varepsilon ,t} \mathrm{d} t+ \beta \varepsilon ^{\frac{d-2} {2}}\, \, u_{\varepsilon , t} \, \mathrm{d} B_{\varepsilon ,t} , \;\;u_{\varepsilon ,0}=1,\] in dimension $d\geq 3$, where $B_{\varepsilon ,t}$ is a spatially smoothed (at scale $\varepsilon $) space-time white noise, and $\beta >0$ is a parameter. We show the existence of a $\bar \beta \in (0,\infty )$ so that the solution exhibits weak disorder when $\beta <\bar \beta $ and strong disorder when $\beta > \bar \beta $. The proof techniques use elements of the theory of the Gaussian multiplicative chaos.

There are three results each concerning large but remote deterministic time intervals at which excursions of a process away from the origin must occur. The first result gives a sufficient condition for a persistent random walk with a finite fourth moment. In this instance the aforementioned time intervals include an additional requirement that the walk is far away from the origin. The second result gives a necessary and a sufficient condition for similar excursions in the case of Brownian motion. The third result gives a necessary and a sufficient condition for time intervals to be free of the zeros of a class of persistent natural scale linear diffusions on the line and is equivalent to the determination of recurrent sets at infinity of the inverse local time.

According to the Dudley-Wichura extension of the Skorohod representation theorem, convergence in distribution to a limit in a separable set is equivalent to the existence of a coupling with elements converging a.s.in the metric. A density analogue of this theorem says that a sequence of probability densities on a general measurable space has a probability density as a pointwise lower limit if and only if there exists a coupling with elements converging a.s.in the discrete metric. In this paper the discrete-metric theorem is extended to stochastic processes considered in a widening time window. The extension is then used to prove the separability version of the Skorohod representation theorem. The paper concludes with an application to Markov chains.

Kang, Perkins, and Spencer [7] conjectured that the size of the largest component of the Bohman-Frieze process at a fixed time $t$ smaller than $t_c$, the critical time for the process, is $L_1(t)=O(\log n/(t_c-t)^2)$ with high probability. Bhamidi, Budhiraja, and Wang [3] have shown that a bound of the form $L_1(t_n)=O((\log n)^4/(t_c-t_n)^2)$ holds with high probability for $t_n\leq t_c-n^{-\gamma }$ where $\gamma \in (0,1/4)$. In this paper, we improve the result in [3] by showing that for any fixed $\lambda >0$, $L_1(t_n)=O(\log n/(t_c-t_n)^2)$ with high probability for $t_n\leq t_c-\lambda n^{-1/3}$. In particular, this settles the conjecture in [7].

The cut pseudo-metric on the space of graph limits induces an equivalence relation. The quotient space obtained by collapsing each equivalence class to a point is a metric space with appealing analytic properties. We show the equivalence relation admits a Borel lifting: There exists a Borel-measurable mapping that maps each equivalence class to one of its elements. The result yields a general framework for proving measurability properties on the space of graph limits. We give several examples, including Borel-measurability of the set of isomorphism classes of random-free graphons.

Markovian growth-fragmentation processes describe a family of particles which can grow larger or smaller with time, and occasionally split in a conservative manner. They were introduced in [3], where special attention was given to the self-similar case. A Malthusian condition was notably given under which the process does not locally explode, in the sense that for all times, the masses of all the particles can be listed in non-increasing order. Our main result in this work states the converse: when this condition is not verified, then the growth-fragmentation process explodes almost surely. Our proof involves using the additive martingale to bias the probability measure and obtain a spine decomposition of the process, as well as properties of self-similar Markov processes.

It is shown that \[ \mathsf{P} (\|a_1U_1+\dots +a_nU_n\|>u)\le c\,\mathsf{P} (a\|Z_d\|>u) \] for all real $u$, where $U_1,\dots ,U_n$ are independent random vectors uniformly distributed on the unit sphere in $\mathbb{R} ^d$, $a_1,\dots ,a_n$ are any real numbers, $a:=\sqrt{(a_1^2+\dots +a_n^2)/d} $, $Z_d$ is a standard normal random vector in $\mathbb{R} ^d$, and $c=2e^3/9=4.46\dots $. This constant factor is about $89$ times as small as the one in a recent result by Nayar and Tkocz, who proved, by a different method, a corresponding conjecture by Oleszkiewicz. As an immediate application, a corresponding upper bound on the tail probabilities for the norm of the sum of arbitrary independent spherically invariant random vectors is given.

On a large finite connected graph let edges $e$ become “open” at independent random Exponential times of arbitrary rates $w_e$. Under minimal assumptions, the time at which a giant component starts to emerge is weakly concentrated around its mean.

Let $X(t)$, $t\geq 0$, be a Lévy process in $\mathbb{R} ^d$ starting at the origin. We study the closed convex hull $Z_s$ of $\{X(t): 0\leq t\leq s\}$. In particular, we provide conditions for the integrability of the intrinsic volumes of the random set $Z_s$ and find explicit expressions for their means in the case of symmetric $\alpha $-stable Lévy processes. If the process is symmetric and each its one-dimensional projection is non-atomic, we establish that the origin a.s. belongs to the interior of $Z_s$ for all $s>0$. Limit theorems for the convex hull of Lévy processes with normal and stable limits are also obtained.

We study the model of binary branching Brownian motion with spatially-inhomogeneous branching rate $\beta \delta _0(\cdot )$, where $\delta _0(\cdot )$ is the Dirac delta function and $\beta $ is some positive constant. We show that the distribution of the rightmost particle centred about $\frac{\beta } {2}t$ converges to a mixture of Gumbel distributions according to a martingale limit. Our results form a natural extension to S. Lalley and T. Sellke [10] for the degenerate case of catalytic branching.

We provide a characterization of the family of non-negative local martingales that have continuous running supremum and vanish at infinity. This is done by describing the class of random times that identify the times of maximum of such processes. In this way we extend to the case of general filtrations a result proved by Nikeghbali and Yor [21] for continuous filtrations. Our generalization is complementary to the one presented by Kardaras [15], and is obtained by means of similar tools.

We discuss a natural game of competition and solve the corresponding mean field game with common noise when agents’ rewards are rank-dependent. We use this solution to provide an approximate Nash equilibrium for the finite player game and obtain the rate of convergence.

We obtain necessary and sufficient conditions for the Marchenko-Pastur theorem for matrices with IID isotropic rows. Our conditions are related to a weak concentration property for certain quadratic forms of the rows.

Among all generalized Ornstein-Uhlenbeck processes which sample the same invariant measure and for which the same amount of randomness (a $N$-dimensional Brownian motion) is injected in the system, we prove that the asymptotic rate of convergence is maximized by a non-reversible hypoelliptic one.

Starting from the hyperbolic Brownian motion as a time-changed Brownian motion, we explore a set of probabilistic models–related to the SABR model in mathematical finance–which can be obtained by geometry-preserving transformations, and show how to translate the properties of the hyperbolic Brownian motion (density, probability mass, drift) to each particular model. Our main result is an explicit expression for the probability of any of these models hitting the boundary of their domains, the proof of which relies on the properties of the aforementioned transformations as well as time-change methods.

This paper constructs a new interacting particle system on $\mathbb{Z} _{\geq 0}\times \mathbb{Z} _+$ with geometric jumps near the boundary $\{0\}\times \mathbb{Z} _+$ which partially reflects the particles. The projection to each horizontal level is Markov, and on every level the dynamics match stochastic matrices constructed from pure alpha characters of $Sp(\infty )$, while on every other level they match an interacting particle system from Pieri formulas for $Sp(2r)$. Using a previously discovered correlation kernel, asymptotics are shown to be the Discrete Jacobi and Symmetric Pearcey processes.

Consider a Markov process $\boldsymbol{\Phi } =\{ \Phi (t) : t\geq 0\}$ evolving on a Polish space $\mathsf{X} $. A version of the $f$-Norm Ergodic Theorem is obtained: Suppose that the process is $\psi $-irreducible and aperiodic. For a given function $f\colon \mathsf{X} \to [1,\infty )$, under suitable conditions on the process the following are equivalent:

(i) There is a unique invariant probability measure $\pi $ satisfying $\int f\,d\pi <\infty $.

(ii) There is a closed set $C$ satisfying $\psi (C)>0$ that is “self $f$-regular.”

(iii) There is a function $V\colon \mathsf{X} \to (0,\infty ]$ that is finite on at least one point in $\mathsf{X} $, for which the following Lyapunov drift condition is satisfied, \[ \mathcal{D} V\leq - f+b\mathbb{I}_C\, , \tag{V3} \] where $C$ is a closed small set and $\mathcal{D}$ is the extended generator of the process.

For discrete-time chains the result is well-known. Moreover, in that case, the ergodicity of $\boldsymbol{\Phi }$ under a suitable norm is also obtained: For each initial condition $x\in \mathsf{X} $ satisfying $V(x)<\infty $, and any function $g\colon \mathsf{X} \to \mathbb{R} $ for which $|g|$ is bounded by $f$, \[ \lim _{t\to \infty } \mathsf{E}_x[g(\Phi (t))] = \int g\,d\pi . \] Possible approaches are explored for establishing appropriate versions of corresponding results in continuous time, under appropriate assumptions on the process $\boldsymbol{\Phi } $ or on the function $g$.

Let $\alpha :[0,1]\to [0,1]$ be a measurable function. It was proved by P. Marchal [2] that the function \[ \phi ^{(\alpha )}(\lambda ):=\exp \left [ \int _0^1\frac{\lambda -1} {1+(\lambda -1)x}\,\alpha (x)\,\mathrm{d} x \right ],\quad \lambda >0 \] is a special Bernstein function. Marchal used this to construct, on a single probability space, a family of regenerative sets $\mathcal R^{(\alpha )}$ such that $\mathcal{R} ^{(\alpha )} \stackrel{\text {law}} {=} \overline{\{S^{(\alpha )}_t:t\geq 0\}} $ ($S^{(\alpha )}$ is the subordinator with Laplace exponent $\phi ^{(\alpha )}$) and $\mathcal R^{(\alpha )}\subset \mathcal R^{(\beta )}$ whenever $\alpha \leq \beta $. We give two simple proofs showing that $\phi ^{(\alpha )}$ is a complete Bernstein function and extend Marchal’s construction to all complete Bernstein functions.

We consider the Feynman-Kac penalization problem for critical measures of symmetric $\alpha $-stable processes via large time asymptotics of Feynman-Kac functionals.

The strong continuity argument in [6] did not identify an appropriate Banach space. We do so here. A corrected version has been uploaded to arxiv.org/abs/1512.07392.

We establish a sufficient condition for the tightness of a sequence of stochastic processes. Our condition makes it possible to study processes with accumulations of fixed times of discontinuity. Our motivation comes from the study of processes in varying or random environment. We demonstrate the usefulness of our condition on two examples: Galton Watson branching processes in varying environment and logistic branching processes with catastrophes.

The frog model on the rooted $d$-ary tree changes from transient to recurrent as the number of frogs per site is increased. We prove that the location of this transition is on the same order as the degree of the tree.

For any $m \geq 2$, the homozygosity of order $m$ of a population is the probability that a sample of size $m$ from the population consists of the same type individuals. Assume that the type proportions follow Kingman’s Poisson-Dirichlet distribution with parameter $\theta $. In this paper we establish the large deviation principle for the naturally scaled homozygosity as $\theta $ tends to infinity. The key step in the proof is a new representation of the homozygosity. This settles an open problem raised in [1]. The result is then generalized to the two-parameter Poisson-Dirichlet distribution.

We prove the Harnack inequality and boundary Harnack principle for the absolute value of a one-dimensional recurrent subordinate Brownian motion killed upon hitting 0, when 0 is regular for itself and the Laplace exponent of the subordinator satisfies certain global scaling conditions. Using the conditional gauge theorem for symmetric Hunt processes we prove that the Green function of this process killed outside of some interval $(a,b)$ is comparable to the Green function of the corresponding killed subordinate Brownian motion. We also consider several properties of the compensated resolvent kernel $h$, which is harmonic for our process on $(0,\infty )$.

We study the mutual regularity properties of Palm measures of point processes, and establish that a key determining factor for these properties is the rigidity-tolerance behaviour of the point process in question (for those processes that exhibit such behaviour). Thereby, we extend the results of [23], [2], [20] to new ensembles, particularly those that are devoid of any determinantal structure. These include the zeroes of the standard planar Gaussian analytic function and several others.

We show that matrix elements of functions of $N\times N$ Wigner matrices fluctuate on a scale of order $N^{-1/2}$ and we identify the limiting fluctuation. Our result holds for any function $f$ of the matrix that has bounded variation thus considerably relaxing the regularity requirement imposed in [7, 11].