# Exponential integrability and exit times of diffusions on sub-Riemannian and metric measure spaces

## TL;DR

This paper establishes moment estimates, exponential integrability, and exit time bounds for diffusions in sub-Riemannian and metric measure spaces, extending classical results beyond Riemannian geometry.

## Contribution

It introduces new probabilistic estimates for diffusions on sub-Riemannian limits and RCD* spaces, broadening the scope of geometric analysis techniques.

## Key findings

- Derived exponential integrability and concentration inequalities for diffusions.
- Established exit time estimates in non-Riemannian settings.
- Obtained Carmona-type eigenfunction estimates for Schrödinger operators.

## Abstract

In this article we derive moment estimates, exponential integrability, concentration inequalities and exit times estimates for canonical diffusions in two settings each beyond the scope of Riemannian geometry. Firstly, we consider sub-Riemannian limits of Riemannian foliations. Secondly, we consider the non-smooth setting of $\mathrm{RCD}^*(K,N)$ spaces. In each case the necessary ingredients are an It\^{o} formula and a comparison theorem for the Laplacian, for which we refer to the recent literature. As an application, we derive pointwise Carmona-type estimates on eigenfunctions of Schr\"{o}dinger operators.