Weak Poincar\'e Inequalities for Convergence Rate of Degenerate Diffusion Processes

TL;DR

This paper extends the study of hypocoercivity by establishing weak Poincaré inequalities to estimate convergence rates of degenerate diffusion processes, providing new insights into non-exponential decay behaviors.

Contribution

It introduces a novel approach using weak Poincaré inequalities for symmetric and anti-symmetric parts of generators to analyze convergence rates in degenerate diffusions.

Findings

Non-exponential convergence rates characterized

Extension of hypocoercivity analysis

Concrete examples demonstrating the theory

Abstract

For a contraction $C_0$-semigroup on a separable Hilbert space, the decay rate is estimated by using the weak Poincar\'e inequalities for the symmetric and anti-symmetric part of the generator. As applications, non-exponential convergence rate is characterized for a class of degenerate diffusion processes, so that the study of hypocoercivity is extended. Concrete examples are presented.