Potential theory of infinite dimensional L\'evy processes

TL;DR

This paper develops potential theory for infinite dimensional Lévy processes, including Brownian motion on abstract Wiener spaces, by constructing compact Lyapunov functions and extending classical potential theory results to this setting.

Contribution

It introduces the construction of compact Lyapunov functions for infinite dimensional Lévy processes, enabling the transfer of classical potential theory techniques to this complex setting.

Findings

Construction of compact Lyapunov functions for infinite dimensional Lévy processes

Extension of potential theoretic principles to infinite dimensions

Solution of the Dirichlet problem with general boundary data in this context

Abstract

We study the potential theory of a large class of infinite dimensional L\'evy processes, including Brownian motion on abstract Wiener spaces. The key result is the construction of compact Lyapunov functions, i.e. excessive functions with compact level sets. Then many techniques from classical potential theory carry over to this infinite dimensional setting. Thus a number of potential theoretic properties and principles can be proved, answering long standing open problems even for the Brownian motion on abstract Wiener space, as e.g. formulated by R. Carmona in 1980. In particular, we prove the analog of the known result, that the Cameron-Martin space is polar, in the L\'evy case and apply the technique of controlled convergence to solve the Dirichlet problem with general (not necessarily continuous) boundary data.