Conservativeness of non-symmetric diffusion processes generated by perturbed divergence forms

TL;DR

This paper establishes criteria for the conservativeness of non-symmetric diffusion processes in unbounded domains, using advanced martingale decomposition techniques to analyze perturbed divergence form operators.

Contribution

It introduces new conservativeness criteria for non-symmetric diffusions associated with general perturbed divergence form operators in unbounded domains.

Findings

Provides criteria for process conservativeness in unbounded domains.

Extends martingale decomposition methods to non-symmetric diffusions.

Connects the approach to the classical Lyons-Zheng decomposition in symmetric cases.

Abstract

Let E be an unbounded open (or closed) domain in Euclidean space of dimension greater or equal to two. We present conservativeness criteria for (possibly reflected) diffusions with state space E that are associated to fairly general perturbed divergence form operators. Our main tool is a recently extended forward and backward martingale decomposition, which reduces to the well-known Lyons-Zheng decomposition in the symmetric case.