On uniqueness of solutions to martingale problems --- counterexamples and sufficient criteria

TL;DR

This paper presents counterexamples where analytic symbols do not uniquely determine Markov process laws and establishes conditions under which smoothness and ellipticity ensure uniqueness.

Contribution

It provides counterexamples of non-uniqueness and introduces sufficient criteria involving smoothness and ellipticity for uniqueness of solutions to martingale problems.

Findings

Analytic symbols can fail to determine Markov process laws uniquely.

Smoothness and ellipticity of the symbol guarantee uniqueness in law.

Polynomial process laws are not always determined by their generators.

Abstract

The dynamics of a Markov process are often specified by its infinitesimal generator or, equivalently, its symbol. This paper contains examples of analytic symbols which do not determine the law of the corresponding Markov process uniquely. These examples also show that the law of a polynomial process is not necessarily determined by its generator. On the other hand, we show that a combination of smoothness of the symbol and ellipticity warrants uniqueness in law. The proof of this result is based on proving stability of univariate marginals relative to some properly chosen distance.