L\'evy processes with values in locally convex Suslin spaces

TL;DR

This paper extends the theory of Lévy processes to complex infinite-dimensional spaces, providing a decomposition and conditions for handling infinite activity in these settings.

Contribution

It introduces a Lévy-Itô decomposition for Lévy processes in complete locally convex Suslin spaces, broadening the scope beyond Banach spaces.

Findings

Provides a Lévy-Itô decomposition in locally convex Suslin spaces

Establishes conditions for the existence of pathwise compensated Poisson integrals

Includes examples like Fréchet and distribution spaces

Abstract

We provide a L\'evy-It\^o decomposition of sample paths of L\'evy processes with values in complete locally convex Suslin spaces. This class of state spaces contains the well investigated examples of separable Banach spaces, as well as Fr\'echet or distribution spaces among many others. Sufficient conditions for the existence of a pathwise compensated Poisson integral handling infinite activity of the L\'evy process are given.