Existence and uniqueness of (infinitesimally) invariant measures for second order partial differential operators on Euclidean space

TL;DR

This paper investigates conditions under which second order elliptic PDEs on Euclidean space have unique invariant measures, linking recurrence properties with measure uniqueness, and providing examples and criteria for these phenomena.

Contribution

It establishes that recurrence ensures the existence and uniqueness of invariant and infinitesimally invariant measures for elliptic operators with low regularity coefficients.

Findings

Recurrence implies uniqueness of infinitesimally invariant measures.

Explicit criteria for recurrence are applicable to measure uniqueness.

Counterexamples show failure of L^1-uniqueness for some measures.

Abstract

We consider a locally uniformly strictly elliptic second order partial differential operator in $\mathbb{R}^d$, $d\ge 2$, with low regularity assumptions on its coefficients, as well as an associated Hunt process and semigroup. The Hunt process is known to solve a corresponding stochastic differential equation that is pathwise unique. In this situation, we study the relation of invariance, infinitesimal invariance, recurrence, transience, conservativeness and $L^r$-uniqueness, and present sufficient conditions for non-existence of finite infinitesimally invariant measures as well as finite invariant measures. Our main result is that recurrence implies uniqueness of infinitesimally invariant measures, as well as existence and uniqueness of invariant measures, both in subclasses of locally finite measures. We can hence make in particular use of various explicit analytic criteria for recurrence that have been previously developed in the context of (generalized) Dirichlet forms and present diverse examples and counterexamples for uniqueness of infinitesimally invariant, as well as invariant measures and an example where $L^1$-uniqueness fails for one infinitesimally invariant measure but holds for another and pathwise uniqueness holds. Furthermore, we illustrate how our results can be applied to related work and vice versa.