On time-dependent functionals of diffusions corresponding to divergence form operators

TL;DR

This paper studies additive functionals of diffusions linked to divergence form operators, establishing their decomposition into martingale and zero-quadratic variation parts, and extending Itô's formula under certain conditions.

Contribution

It introduces conditions under which additive functionals of nonhomogeneous diffusions are continuous Dirichlet processes and extends Itô's formula for these functionals.

Findings

Additive functionals are continuous Dirichlet processes for almost every starting point.

Decomposition into martingale and zero-quadratic variation parts is characterized.

Itô's formula is extended for these functionals under specified conditions.

Abstract

We consider additive functionals as a time and space-dependent function of a diffusion corresponding to nonhomogeneous uniformly elliptic divergence form operator. We show that if the function belongs to natural domain of strong solutions of PDEs then there is a version of this function such that additive functional is a continuous Dirichlet process for almost every starting points of diffusion and we describe the martingale and the zero-quadratic variation parts of its decomposition. We show also that under slightly stronger condition on the function this property holds for every starting point. Finally, we prove an extension of the It\^o formula.