Analysis and stochastic processes on metric measure spaces

Abstract

This contribution deals with the properties of certain differential and nonlocal operators on various spaces, with the emphasis on the relationship between the analytic properties of the operators in question and the geometric properties of the underlying space. In most situations, these operators are Markov generators. In such cases, we are also concerned with probabilistic aspects, such as the path properties of the corresponding Markov process. Analysis on manifoldsThe main object of interest in this part of the project is the Laplace-Beltrami operator Δ on a Riemannian manifold M . In most cases M can be assumed geodesically complete and non-compact. Denote by B (x, r) the geodesic ball on M of radius r centered at x ∈ M , and by V (x, r) the Riemannian volume of B (x, r).Manifold M is called parabolic if any positive superharmonic function on M is const. It is known that the following properties are equivalent:• M is parabolic;• there is no positive Green function of Δ on M ;• Brownian motion on M is recurrent;(see [16]).