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

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.(c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This article discusses Fokker-Planck equations driven by Levy processes in the entire Euclidean space, under the influence of confining drifts, similar to the classical Ornstein-Ulhenbeck model. A new PDE method is introduced to obtain exponential or sub-exponential decay rates of zero average solutions as time goes to infinity, under certain diffusivity conditions on the Levy process, including the fractional Laplace operator as a model example. The approach relies on long-time oscillation estimates of the adjoint problem and applies to both local and nonlocal diffusions, as well as strongly or weakly confining drifts.