Marina Vachkovskaia

Survival of Branching Random Walks in Random Environment

We study survival of nearest-neighbor branching random walks in random environment (BRWRE) on ℤ. A priori there are three different regimes of survival: global survival, local survival, and strong local survival. We show that local and strong local survival regimes coincide for BRWRE and that they can be characterized with the spectral radius of the first moment matrix of the process. These results are generalizations of the classification of BRWRE in recurrent and transient regimes. Our main result is a characterization of global survival that is given in terms of Lyapunov exponents of an infinite product of i.i.d. 2×2 random matrices.

Billiards in a General Domain with Random Reflections

We study stochastic billiards on general tables: a particle moves according to its constant velocity inside some domain

On the connectivity properties of the complementary set in fractal percolation models

\fancyscript{D}\subset {\mathbb{R}}^d

Knudsen Gas in a Finite Random Tube: Transport Diffusion and First Passage Properties

until it hits the boundary and bounces randomly inside, according to some reflection law. We assume that the boundary of the domain is locally Lipschitz and almost everywhere continuously differentiable. The angle of the outgoing velocity with the inner normal vector has a specified, absolutely continuous density. We construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location/velocity. We mainly focus on the case of bounded domains. Then, we prove exponential ergodicity of these two Markov processes, we study their invariant distribution and their normal (Gaussian) fluctuations. Of particular interest is the case of the cosine reflection law: the stationary distributions for the two processes are uniform in this case, the discrete time chain is reversible though the continuous time process is quasi-reversible. Also in this case, we give a natural construction of a chord “picked at random” in

Quenched invariance principle for the Knudsen stochastic billiard in a random tube

\fancyscript{D}

Two-Dimensional Random Interlacements and Late Points for Random Walks

, and we study the angle of intersection of the process with a (d − 1)-dimensional manifold contained in