On convergence of 1D Markov diffusions to heavy-tailed invariant density

TL;DR

This paper investigates the convergence rates of a one-dimensional diffusion process with heavy-tailed invariant distribution, demonstrating how to construct a process that converges exponentially fast uniformly from any initial state.

Contribution

It introduces a method to modify diffusion processes to achieve exponential convergence to heavy-tailed invariant measures, improving upon polynomial convergence guarantees.

Findings

Constructed a diffusion process with exponential convergence rate

Achieved uniform convergence from all initial conditions

Extended polynomial convergence results to exponential rates

Abstract

Rate of convergence is studied for a diffusion process on the half line with a non-sticky reflection to a heavy-tailed 1D invariant distribution which density on the half line has a polynomial decay at infinity. Starting from a standard receipt which guarantees some polynomial convergence, it is shown how to construct a new non-degenerate diffusion process on the half line which converges to the same invariant measure exponentially fast uniformly with respect to the initial data.