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Short and long time behavior of the Fokker-Planck equation in a confining potential and applications

Frederic Herau

TL;DR

The paper analyzes the linear Fokker-Planck equation in a confining potential for dimensions $d\ge 3$, establishing sharp short-time smoothing and long-time exponential decay bounds via a hypocoercivity framework. It develops a robust functional-analytic setup with the Maxwellian weight, Witten Laplacian, and Sobolev scales, and proves precise derivative bounds for the FP semigroup. Under a spectral-gap assumption, it proves exponential convergence to equilibrium and extends the results to nonlinear, mollified VPFP systems, including global existence of solutions and exponential decay with entropy-type estimates. The work provides a methodological template for combining hypoelliptic diffusion, spectral gaps, and nonlinear mollified couplings in kinetic equations.

Abstract

We consider the linear Fokker-Planck equation in a confining potential in space dimension $d \geq 3$. Using spectral methods, we prove bounds on the derivatives of the solution for short and long time, and give some applications.

Short and long time behavior of the Fokker-Planck equation in a confining potential and applications

TL;DR

The paper analyzes the linear Fokker-Planck equation in a confining potential for dimensions , establishing sharp short-time smoothing and long-time exponential decay bounds via a hypocoercivity framework. It develops a robust functional-analytic setup with the Maxwellian weight, Witten Laplacian, and Sobolev scales, and proves precise derivative bounds for the FP semigroup. Under a spectral-gap assumption, it proves exponential convergence to equilibrium and extends the results to nonlinear, mollified VPFP systems, including global existence of solutions and exponential decay with entropy-type estimates. The work provides a methodological template for combining hypoelliptic diffusion, spectral gaps, and nonlinear mollified couplings in kinetic equations.

Abstract

We consider the linear Fokker-Planck equation in a confining potential in space dimension . Using spectral methods, we prove bounds on the derivatives of the solution for short and long time, and give some applications.

Paper Structure

This paper contains 10 sections, 13 theorems, 126 equations.

Table of Contents

  1. Introduction and results
  2. Functional analysis
  3. Short time behavior
  4. Exponential time decay
  5. An abstract Hilbert result
  6. The case of the Fokker-Planck operator
  7. Applications
  8. Strong solutions for a given interaction potential
  9. A mollified Vlasov-Poisson-Fokker-Planck system
  10. Exponential time decay for small nonlinear coupling

Key Result

Theorem 1.1

There exists a constant C such that for all t >0, we have the following: i) (-\partial_v + v)e^{-tK} \textrm{ is bounded by } C (1+t^{-1/2}) and ii) (-\partial_x + \partial_x V) e^{-tK} \textrm{ is bounded by } C (1+t^{-3/2}), as bounded operators on B^2. Here C depends only on \left\Vert V"\right

Theorems & Definitions (13)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Corollary 1.4
  • Proposition 2.1: HelN04-HN04
  • Proposition 2.2: HN04-HelN04
  • Proposition 3.1
  • Lemma 4.1
  • Proposition 4.2
  • Proposition 4.5
  • ...and 3 more