We discuss the volume fraction of a model of non--overlapping convex grains. It is obtained from thinning a Poisson process where each point has a weight and is the centre of a grain, by removing any grain that is overlapped…

We investigate excited random walks on $Z^d, d\ge 1,$ and on planar strips $Z\times{0,1,\ldots,L-1}$ which have a drift in a given direction. The strength of the drift may depend on a random i.i.d. environment and on the local time of…

We consider a zero sum stochastic differential game which involves two players, emph{the controller} and emph{the stopper}. The stopper selects the stopping rule which halts the game. The controller chooses the diffusion coefficient of the corresponding state process which is…

In this note we use the boundary classification of diffusions
in order to derive
a criterion for the convergence of perpetual integral functionals
of transient real-valued diffusions.
We present a second approach, based on Khas'minskii's lemma,…

When is a nonlinear filter stable with respect to its initial condition? In spite of the recent progress, this question still lacks a complete answer in general. Currently available results indicate that stability of the filter depends on the signal…

We consider a first-passage percolation (FPP) model on a Delaunay triangulation $\mathcal{D}$ of the plane. In this model each edge $\mathbf{e}$ of $\mathcal{D}$ is independently equipped with a nonnegative random variable $\tau_\mathbf{e}$, with distribution function $\mathbb{F}$, which is interpreted as…

We consider global properties of a codimension one manifold embedded in Euclidean space, as it evolves under an isotropic and volume preserving Brownian flow of diffeomorphisms. In particular, we obtain expressions describing the expected rate of growth of the Lipschitz-Killing…

We prove a finite volume lower bound of the order $\sqrt{\log N}$ on the delocalization of a disordered continuous spin model (resp. effective interface model) in $d=2$ in a box of size $N$. The interaction is assumed to be massless,…

Exponential tail bounds are derived for solutions of max-recursive equations and for max-recursive random sequences, which typically arise as functionals of recursive structures, of random trees or in recursive algorithms. In particular they arise in the worst case analysis of…

In a forest-fire model, each site of the square lattice is either vacant or occupied by a tree. Vacant sites get occupied according to independent rate 1 Poisson processes. Independently at each site ignition occurs according to independent rate lambda…