Convergence Rate for Degenerate Partial and Stochastic Differential Equations via weak Poincar\'e Inequalities

TL;DR

This paper analyzes the long-term behavior of degenerate Kolmogorov operators using weak hypocoercivity and Poincaré inequalities, providing convergence rate estimates for solutions to related PDEs and SDEs.

Contribution

It extends previous results by proving essential m-dissipativity and applying weak Poincaré inequalities to derive convergence rates for degenerate equations.

Findings

Establishes m-dissipativity of degenerate Kolmogorov operators.

Provides $L^2$-convergence rate estimates for solutions.

Demonstrates sub-exponential convergence for degenerate Fokker-Planck equations.

Abstract

We employ weak hypocoercivity methods to study the long-term behavior of operator semigroups generated by degenerate Kolmogorov operators with variable second-order coefficients, which solve the associated abstract Cauchy problem. We prove essential m-dissipativity of the operator, which extends previous results and is key to the rigorous analysis required. We give estimates for the $L^2$-convergence rate by using weak Poincar\'e inequalities. As an application, we obtain estimates for the (sub-)exponential convergence rate of solutions to the corresponding degenerate Fokker-Planck equations and of weak solutions to the corresponding degenerate stochastic differential equation with multiplicative noise.