We consider one-dimensional stochastic differential equations with a boundary condition, driven by a Poisson process. We study existence and uniqueness of solutions and the absolute continuity of the law of the solution. In the case when the coefficients are linear,…

We give two algorithms that allow to get arbitrary precision asymptotics for the harmonic potential of a random walk.

In this paper a necesary and sufficient condition is established for the probability that a Lévy process is positive at time $t$ to tend to 1 as $t$ tends to 0. This condition is expressed in terms of the characteristics…

We study the models of competition and spreading of infection for infinite systems of independent random walks. For the competition model, we investigate the question whether one of the spins prevails with probability one. For the infection spreading, we give…

Proofs and two applications of two general results are given concerning the problem of establishing interior smoothness of probabilistic solutions of elliptic degenerate equations.

We prove well-posedness of the martingale problem for an infinite-dimensional degenerate elliptic operator under appropriate Hölder continuity conditions on the coefficients. These martingale problems include large population limits of branching particle systems on a countable state space in which the…

We define and study a family of Markov processes with state space the compact set of all partitions of $N$ that we call exchangeable fragmentation-coalescence processes. They can be viewed as a combination of homogeneous fragmentation as defined by Bertoin…

We discuss the relationship between one-dimensional Gibbs measures and discrete-time processes (chains). We consider finite-alphabet (finite-spin) systems, possibly with a grammar (exclusion rule). We establish conditions for a stochastic process to define a Gibbs measure and vice versa. Our conditions…

We use Girsanov's theorem to establish a conjecture of Khoshnevisan, Xiao and Zhong that $\phi(r) = r^{N-d/2} (\log \log (\frac{1}{r}))^{d/2}$ is the exact Hausdorff measure function for the zero level set of an $N$-parameter $d$-dimensional additive Brownian motion. We extend…

Let $X_1, \ldots , X_n$ be independent, integer valued random variables, with $p^{\text{th}}$ moments, $p >2$, and let $W$ denote their sum. We prove bounds analogous to the classical non-uniform estimates of the error in the central limit theorem, but…