A note on the uniform ergodicity of diffusion processes
Nikola Sandrić
TL;DR
This paper derives explicit Lyapunov-type conditions for the uniform ergodicity of Itô diffusions in total variation, via the Foster–Lyapunov framework. By verifying open-set irreducibility and aperiodicity and constructing a Lyapunov function from integral quantities $I_{x_0}$ and $\gamma_{x_0}$, it proves exponential convergence to a unique invariant measure under the condition $\Lambda<\infty$. The results apply to a broad class of diffusions and extend to subordinate processes, with practical examples illustrating the conditions and demonstrating preservation of uniform ergodicity under subordination. The work provides a concrete, verifiable set of criteria that complements existing coupling-based approaches and broadens the applicability to diffusion models with nontrivial drift and diffusion structures.
Abstract
In this note, we discuss the uniform ergodicity of a diffusion process given by an Itô stochastic differential equation. We present an integral condition in terms of the drift and diffusion coefficients that ensures the uniform ergodicity of the corresponding transition kernel with respect to the total variation distance. Applications of the obtained results to a class of subordinate diffusion processes are also presented.