Random walks on discrete groups of polynomial volume growth

Abstract

Let µ be a probability measure with finite support on a discrete group of polynomial volume growth. The main purpose of this paper is to study the asymptotic behavior of the convolution powers µ * n of µ. If µ is centered, then we prove upper and lower Gaussian estimates. We prove a central limit theorem and we give a generalization of the Berry-Esseen theorem. These results also extend to noncentered probability measures. We study the associated Riesz transform operators. The main tool is a parabolic Harnack inequality for centered probability measures which is proved by using ideas from homogenization theory and by adapting the method of Krylov and Safonov. This inequality implies that the positive µ-harmonic functions are constant. Finally we give a characterization of the µ-harmonic functions which grow polynomially.

Lévy area with a drift as a renormalization limit of Markov chains on periodic graphs

“…A result due to Wehn (see, for example, [7], theorem 1.3, for details) states that, when is a centered probability measure on a connected Lie group, * (the th convolution of ) converges to the Wiener measure (under certain conditions on ). In [1], the main result states that, when is a probability measure with finite support on a discrete group of polynomial volume growth (nilpotent Lie groups, and in particular (R ), are of polynomial volume growth), * converges to the heat kernel of a centered left-invariant sub-Laplacian on a certain simply connected nilpotent Lie group. In both cases, we deal with i.i.d.…”

“…increments and no area anomaly is exhibited at the limit. However, in [1] the possibility of a non-centered measure * is taken into account, just to show that the asymptotic behavior is similar to the non-centered case modulo a transformation by a multiplicative function (which is equivalent to re-centring ). What this shows in particular is that our area anomaly is not a question of the process drift but a new phenomenon.…”

“…In [2], the main result from [1] is generalized to sub-Laplacians with drift (i.e. we are here in a continuous setting).…”

“…we are here in a continuous setting). Just like in [1], the author gets a Berry-Esseen-like estimate of the heat kernel. The drift of the Laplacian is not automatically linked with the area anomaly and is considered here to be more like an additional problem than a central object that should be studied.…”

“…. , ), the second level is determined by the increments of the process (the first level) and the stochastic area: a process on the space of × antisymmetric matrices given by (1) , ( ) = (︂∫︁ ∫︁…”