Gradient flow formulation and longtime behaviour of a constrained Fokker-Planck equation

TL;DR

This paper analyzes a constrained Fokker-Planck equation as a Wasserstein gradient flow, proving existence of solutions and providing explicit convergence rates to equilibrium under evolving constraints.

Contribution

It introduces a novel interpretation of the constrained Fokker-Planck equation as a Wasserstein gradient flow and establishes existence and convergence results.

Findings

Existence of solutions via an explicit Euler scheme.

Quantitative convergence rates to equilibrium.

Explicit dependence of convergence on logarithmic Sobolev constants.

Abstract

We consider a Fokker-Planck equation which is coupled to an externally given time-dependent constraint on its first moment. This constraint introduces a Lagrange-multiplier which renders the equation nonlocal and nonlinear. In this paper we exploit an interpretation of this equation as a Wasserstein gradient flow of a free energy ${\mathcal{F}}$ on a time-constrained manifold. First, we prove existence of solutions by passing to the limit in an explicit Euler scheme obtained by minimizing $h {\mathcal{F}}(\varrho)+W_2^2(\varrho^0,\varrho)$ among all $\varrho$ satisfying the constraint for some $\varrho^0$ and time-step $h>0$. Second, we provide quantitative estimates for the rate of convergence to equilibrium when the constraint converges to a constant. The proof is based on the investigation of a suitable relative entropy with respect to minimizers of the free energy chosen according to the constraint. The rate of convergence can be explicitly expressed in terms of constants in suitable logarithmic Sobolev inequalities.