Harnack Inequality and Applications for Infinite-Dimensional GEM Processes

TL;DR

This paper establishes a dimension-free Harnack inequality and heat kernel bounds for infinite-dimensional GEM processes, enhancing understanding of their probabilistic and functional inequalities.

Contribution

It introduces new Harnack and super log-Sobolev inequalities for infinite-dimensional GEM processes, extending previous results and employing coupling techniques.

Findings

Dimension-free Harnack inequality derived

Uniform heat kernel bounds established

Super log-Sobolev inequality proven

Abstract

The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in \cite{FW} to simulate the two-parameter GEM distributions. In particular, the associated Dirichlet form satisfies the super log-Sobolev inequality which strengthens the log-Sobolev inequality derived in \cite{FW}. To prove the main results, explicit Harnack inequality and super Poincar\'e inequality are established for the one-dimensional Wright-Fisher diffusion processes. The main tool of the study is the coupling by change of measures.