Let (Zn)n Î N0 be a d-dimensional random walk in random scenery, i.e., Zn = åk=0n-1 Y(Sk) with (Sk)k Î N0 a random walk in Zd and ( Y(z) )z Î Zd an i.i.d. scenery, independent of the walk. The walker's steps have mean zero and finite variance.
We identify the speed and the rate of the logarithmic decay of P([ 1/(n)] Zn > bn) for various choices of sequences (bn)n in [1,¥). Depending on (bn)n and the upper tails of the scenery, we identify different regimes for the speed of decay and different variational formulas for the rate functions. In contrast to recent work by A. Asselah and F. Castell, we consider sceneries unbounded to infinity. It turns out that there are interesting connections to large deviation properties of self-intersections of the walk, which have been studied recently by X. Chen.
Reinforcement is observed frequently in nature and society, where
beneficial interactions tend to be repeated. Edge reinforced random
walker on a graph remembers the number of times each edge was
traversed in the past, and decides to make the next random step with
probabilities favouring places visited before. Using martingale
techniques and comparison with the generalized Urn scheme, it is shown
in  that the edge reinforced random walker on a graph of bounded
degree, with the reinforcement weight function W(k) = kr, r > 1,
traverses a random attracting edge at all large times, with
probability 1. A remarkably short argument of Sellke  shows that
attracting edge exists if and only if
Progress has been made recently  towards better understanding of attracting edge property for convex and increasing weights W with property (1).
A reversible measure turns a Markov operator into a symmetric operator and spectral theory can then be used to study the transition kernel. We introduce the more general notion of `centered measure' and prove an upper estimate on the decay of the transition probabilities of Carne-Varopoulos type. The connection with the rate of escape in the case of random walks on discrete groups will be discussed.
A self-interacting random walk is a random process evolving in an environment depending on its past behavior.
The notion of Edge-Reinforced Random Walk (ERRW) was introduced in 1986 by Coppersmith and Diaconis  on a discrete graph, with the probability of a move along an edge being proportional to the number of visits to this edge. In the same spirit, Pemantle introduced in 1988  the Vertex-Reinforced Random Walk (VRRW), the probability of a move to an adjacent vertex being then proportional to the number of visits to this vertex (and not to the edge leading to the vertex). The Self-Interacting Diffusion (SID) is a continuous counterpart to these notions.
Although introduced by similar definitions, these processes show some significantly different behaviors, leading in their understanding to various methods. While the study of ERRW essentially requires some probabilistic tools, corresponding to some local properties, the comprehension of VRRW and SID needs a joint understanding of on one hand a dynamical system governing the general evolution, and on the other hand some probabilistic phenomena, acting as a perturbation, and sometimes changing the nature of this dynamical system.
The purpose of our talk is to present our recent results on the subject , , , .