On the construction of convolution-like operators associated with multidimensional diffusion processes

TL;DR

This paper investigates when multidimensional diffusion processes can be associated with convolution-like operators, providing necessary and sufficient conditions and exploring specific cases like reflected Brownian motions on bounded domains and Riemannian manifolds.

Contribution

It systematically characterizes the existence of convolution-like operators for general strong Feller processes, extending previous one-dimensional results to higher dimensions and complex geometries.

Findings

Necessary and sufficient conditions for convolution-like structures are established.

Connections between eigenfunctions of transition semigroups and convolution structures are identified.

Analysis of reflected Brownian motions on bounded domains and Riemannian manifolds is provided.

Abstract

When is it possible to interpret a given Markov process as a L\'evy-like process? Since the class of L\'evy processes can be defined by the relation between transition probabilities and convolutions, the answer to this question lies in the existence of a convolution-like operator satisfying the same relation with the transition probabilities of the process. It is known that the so-called Sturm-Liouville convolutions have the desired properties and therefore the question above has a positive answer for a certain class of one-dimensional diffusions. However, more general processes have never been systematically treated in the literature. This study addresses this gap by considering the general problem of constructing a convolution-like operator for a given strong Feller process on a general locally compact metric space. Both necessary and sufficient conditions for the existence of such convolution-like structures are determined, which reveal a connection between the answer to the above question and certain analytical and geometrical properties of the eigenfunctions of the transition semigroup. The case of reflected Brownian motions on bounded domains of R d and compact Riemannian manifolds is considered in greater detail: various special cases are analysed, and a general discussion on the existence of appropriate convolution-like structures is presented.