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 consider a fractional Brownian motion with Hurst parameter strictly between 0 and 1. We are interested in the asymptotic behaviour of functionals of the increments of this and related processes and we propose several probabilistic and statistical applications.

Stochastic partial differential equations of divergence form with discontinuous and unbounded coefficients are considered in $C^1$ domains. Existence and uniqueness results are given in weighted $L_p$ spaces, and Holder type estimates are presented.

The Green function of a transient symmetric Markov process can be interpreted as the covariance of a centered Gaussian process. This relation leads to several fruitful identities in law. Symmetric Markov processes and their associated Gaussian process both benefit from…

If $P_t$ is the semigroup associated with the Kawasaki dynamics on $Z^d$ and $f$ is a local function on the configuration space, then the variance with respect to the invariant measure $\mu$ of $P_t f$ goes to zero as $t\to…

We bound the probability that a continuous time simple random walk on the infinite percolation cluster on $Z^d$ returns to the origin at time $t$. We use this result to show that in dimensions 5 and higher the uniform spanning…

We consider the time for a critical nearest particle system, starting in equilibrium subject to possessing a large gap, to achieve equilibrium.