# Existence and regularity of infinitesimally invariant measures, transition functions and time homogeneous It\^o-SDEs

## TL;DR

This paper establishes the existence and regularity of invariant measures for certain elliptic PDEs and connects these measures to solutions of time-homogeneous Itô SDEs, enabling analysis of their long-term behavior.

## Contribution

It introduces new conditions under which invariant measures exist for elliptic operators and links these measures to Hunt processes solving SDEs, with detailed regularity and ergodic properties.

## Key findings

- Existence of an infinitesimally invariant measure for a broad class of elliptic operators.
- Regularity properties of the associated semigroup, including strong Feller and irreducibility.
- Identification of the transition function as a solution to an SDE with applications to process properties.

## Abstract

We show existence of an infinitesimally invariant measure $m$ for a large class of divergence and non-divergence form elliptic second order partial differential operators with locally Sobolev regular diffusion coefficient and drift of some local integrability order. Subsequently, we derive regularity properties of the corresponding semigroup which is defined in $L^s(\mathbb{R}^d,m)$, $s\in [1,\infty]$, including the classical strong Feller property and classical irreducibility. This leads to a transition function of a Hunt process that is explicitly identified as a solution to an SDE. Further properties of this Hunt process, like non-explosion, moment inequalities, recurrence and transience, as well as ergodicity, including invariance and uniqueness of $m$, and uniqueness in law, can then be studied using the derived analytical tools and tools from generalized Dirichlet form theory.