Decay rates of convergence for Fokker-Planck equations with confining drift

TL;DR

This paper develops a PDE-based approach to analyze the exponential and sub-exponential decay rates of solutions to Fokker-Planck equations driven by Levy processes with confining drifts, unifying and extending previous methods.

Contribution

It introduces a new PDE method for decay rate analysis of Fokker-Planck equations with Levy noise, connecting PDE and probabilistic techniques.

Findings

Established decay rates for solutions with Levy-driven Fokker-Planck equations.

Unified framework encompassing local and nonlocal diffusions.

Extended previous probabilistic and analytic results to broader settings.

Abstract

We consider Fokker-Planck equations in the whole Euclidean space, driven by Levy processes, under the action of confining drifts, as in the classical Ornstein-Ulhenbeck model. We introduce a new PDE method to get exponential or sub-exponential decay rates, as time goes to infinity, of zero average solutions, under some diffusivity condition on the Levy process, which includes the fractional Laplace operator as a model example. Our approach relies on the long time oscillation estimates of the adjoint problem and applies to (the possible superposition of) both local and nonlocal diffusions, as well as to strongly or weakly confining drifts. Our results extend, with a unifying perspective, many previous works based on different analytic or probabilistic methods, with several interesting connections. On one hand, we make a link between the (nonlinear) PDE methods used for the long time behavior of Hamilton-Jacobi equations and the decay estimates of Fokker-Planck equations; on another hand, we give a purely analytical approach towards some oscillation decay estimates which were obtained so far only with probabilistic coupling methods.