Alexander Bendikov

On- and off-diagonal heat kernel behaviors on certain infinite dimensional local Dirichlet spaces

This paper studies on-diagonal and off-diagonal bounds for symmetric diffusion semi-groups that admit a continuous kernel, in the case where the underlying space is typically infinite dimensional. For invariant diffusions on the infinite dimensional torus, the equivalence between a certain on-diagonal estimate and a certain off-diagonal behavior is proved. This gives a necessary and sufficient condition, in terms of the associated infinite symmetric matrix, for an elliptic Harnack inequality to be satisfied. As a consequence, elliptic and parabolic Harnack inequalities are in fact equivalent properties in this setting. In terms of potential theory, this gives a necessary and sufficient condition for the associated harmonic sheaf to be a Brelot harmonic sheaf. A new distance associated to certain Dirichlet spaces is also introduced. It plays a crucial role in relating on-diagonal behavior to off-diagonal behavior in cases where the intrinsic distance is infinite almost everywhere.

Random walks driven by low moment measures

We study the decay of convolution powers of probability measures without second moment but satisfying some weaker finite moment condition. For any locally compact unimodular group G and any positive function ϱ:G→[0,+∞], we introduce a function ΦG,ϱ which describes the fastest possible decay of n↦ϕ(2n)(e) when ϕ is a symmetric continuous probability density such that ∫ϱϕ is finite. We estimate ΦG,ϱ for a variety of groups G and functions ϱ. When ϱ is of the form ϱ=ρ∘δ with ρ:[0,+∞)→[0,+∞), a fixed increasing function, and δ:G→[0,+∞), a natural word length measuring the distance to the identity element in G, ΦG,ϱ can be thought of as a group invariant.

α-stable random walk has massive thorns

We introduce and study a class of random walks defined on the integer lattice

On the spectrum of the hierarchical Laplacian

\mathbb{Z} ^d

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

-- a discrete space and time counterpart of the symmetric

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

\alpha

Some Spectral and Geometric Aspects of Countable Groups

-stable process in

On massive sets for subordinated random walks

\mathbb{R} ^d

Hypoelliptic Bi-Invariant Laplacians on Infinite Dimensional Compact Groups

. When

Isotropic Markov semigroups on ultra-metric spaces

0< \alpha <2