Christophe Pittet

Spectral properties of a class of random walks on locally finite groups

We study some spectral properties of random walks on innite countable amenable groups with an emphasis on loc ally �nite groups, e.g. the innite symmetric group S1 . On locally �nite groups, the random walks under consideration are driv en by innite divisible distributions. This allows us to embed ou r random walks into continuous time Lprocesses whose heat kernels have shapes similar to the ones of �-stable processes. We obtain exam- ples of fast/slow decays of return probabilities, a recurrence crite- rion, exact values and estimates of isospectral proles and spectral distributions, formulae and estimates for the escape rates and for heat kernels.

Random walks on abelian-by-cyclic groups

We describe the large time asymptotic behaviors of the probabilities p 2t (e, e) of return to the origin associated to finite symmetric generating sets of abelian-by-cyclic groups. We characterize the different asymptotic behaviors by simple algebraic properties of the groups.

Изотропные марковские полугруппы на ультраметрических пространствах@@@Isotropic Markov semigroups on ultra-metric spaces

Let (X,d) be a locally compact separable ultra-metric space. Given a reference measure \mu\ on X and a step length distribution on the non-negative reals, we construct a symmetric Markov semigroup P^t acting in L^2(X,\mu). We study the corresponding Markov process. We obtain upper and lower bounds of its transition density and its Green function, give a transience criterion, estimate its moments and describe the Markov generator and its spectrum, which is pure point. In the particular case when X is the field of p-adic numbers, our construction recovers fractional derivative and the Taibleson Laplacian (spectral multiplier), and we can also apply our theory to the study of the Vladimirov Laplacian which is closely related to the concept of p-adic Quantum Mechanics. Even in this well established setting, several of our results are new. We also elaborate the relation between our processes and Kigamis jump processes on the boundary of a tree which are induced by a random walk. In conclusion, we provide examples illustrating the interplay between the fractional derivatives and random walks.

Some Spectral and Geometric Aspects of Countable Groups

We discuss the relationship between the isospectral profile and the spectral distribution of a Laplace operator on a countable group. In the case of locally finite countable groups, we emphasize the relevance of the metric associated to a natural Markov operator: it is an ultra-metric whose balls are optimal sets for the isospectral profile.

Systoles on S1 × Sn

Abstract There is no intersystolic inequality syst 1 syst n ⩽, const Vol on S 1 × S n if n is greater or equal to two.