# Regularity and Stability of Invariant Measures for Diffusion Processes under Synthetic Lower Ricci Curvature Bounds

## TL;DR

This paper establishes Sobolev regularity of invariant measures for diffusion processes on non-smooth spaces with synthetic Ricci bounds, and analyzes their stability and symmetrizability under perturbations.

## Contribution

It proves Sobolev regularity of invariant measures on non-smooth metric measure spaces with Ricci bounds and characterizes semigroup symmetrizability and measure stability.

## Key findings

- Invariant measures have Sobolev regularity under synthetic Ricci bounds.
- Semigroups are characterized by their symmetrizability.
- Invariant measures are stable under perturbations of drifts and space convergence.

## Abstract

The Sobolev regularity of invariant measures for diffusion processes is proved on non-smooth metric measure spaces with synthetic lower Ricci curvature bounds. As an application, the symmetrizability of semigroups is characterized, and the stability of invariant measures is proved under perturbations of drifts and the underlying spaces in the sense of the measured Gromov convergence.