Ergodic Density Estimates for some diffusion processes

TL;DR

This paper establishes uniform upper bounds for the transition and ergodic densities of certain n-dimensional ergodic diffusion processes in the positive orthant, without assuming geodesic completeness of the elliptic symbol.

Contribution

It provides novel time-independent density bounds for ergodic diffusions on the positive orthant without requiring geodesic completeness.

Findings

Derived time-independent upper bounds for transition densities.

Established bounds for the unique ergodic density.

Applicable to diffusion processes with boundary considerations.

Abstract

For n-dimensional ergodic diffusion processes with values in $G=\mathbb{R}_{+}^n$ we prove time-independent upper bounds for the transitional density and so also for the unique ergodic density. We do not require geodesic completeness of the elliptic symbol towards the boundary of $G$.