My current research concerns a few models of random walk, including random walk in random and non-random environments (for example, Sinai's walk), systems of random walks with coalescence, branching random walk, multi-parameter random walk, random environment and random scenery. I am particularly curious about various asymptotic properties of these random walks.

I also get interested in some continuous time processes, including Brownian motion in a random potential (which is closely related to random walk in random environment, though how "closely related" is not clear to me for the moment and I am working hard on this topic; details in this paper), or some special types of Gaussian processes such as fractional Brownian motion (which is closely related to symmetric stable processes via a Ray-Knight theorem; details in this paper).

Small deviations for stochastic processes are connected to topics in other mathematical branches (integral operators, metric entropy, functional quantization, etc). I am currently working actively on such problems, especially for fractional and/or Lévy processes. More information on this site.