A trajectorial approach to the gradient flow of McKean-Vlasov SDEs with mobility

TL;DR

This work develops a trajectorial gradient-flow framework for McKean-Vlasov SDEs with density-dependent drift and diffusion by introducing a modified Wasserstein metric $W_h$ with $h(r)=rb(r)$. It defines a generalized internal energy $E_g$ and free energy $\mathcal{F}$, and proves that the nonlinear Fokker-Planck equation is the $W_h$-gradient flow of $\mathcal{F}$, with energy dissipation governed by a modified Fisher information $I_g$. The authors derive trajectorial entropy dissipation identities and the corresponding Wasserstein-slope formulas, linking macroscopic dissipation to pathwise dynamics and enabling a probabilistic interpretation of gradient flow. They illustrate the theory on the Fermi-Dirac-Fokker-Planck model, demonstrating dissipation rates and gradient-flow structure, and discuss condensation phenomena and non-exponential convergence as directions for future research, highlighting potential extensions to broader mobility regimes and perturbations of the potential.

Abstract

We establish the gradient flow representation of diffusion with mobility $b$ with respect to the modified Wasserstein quasi-metric $W_h$, where $h(r)=rb(r)$. The appropriate selection of the free energy functional depends on the specific form of the generalized entropy. Different from the JKO scheme, we derive the trajectorial version of the relative entropy dissipation identity for the McKean-Vlasov stochastic differential equation (SDE) with Nemytskii-type coefficients, utilizing techniques from stochastic analysis. Based on this, we demonstrate that the trajectorial average of the solution process to the McKean-Vlasov SDE, with respect to the underlying measure, corresponds to the rate of dissipation of the free energy. As an application, we present the energy dissipation of the Fermi-Dirac-Fokker-Planck equation, a model widely used in physics and biology to describe saturation effects. Inspired by numerical simulations, we propose two questions on condensation phenomena and non-exponential convergence rate.

Figures (10)

• Figure 1: The energy $\mathcal{F}_i(p_t)-\mathcal{F}_i(p_0)$ of 500 trajectories.
• Figure 2: The energy $\overline{\mathcal{F}}(p_t)-\overline{\mathcal{F}}(p_0)$ of \ref{['FD']}.
• Figure 3: For $\gamma=1$, the energy $\mathcal{F}_i(p_t)-\mathcal{F}_i(p_0)$ of 500 trajectories.
• Figure 4: For $\gamma=1$, the energy $\overline{\mathcal{F}}(p_t)-\overline{\mathcal{F}}(p_0)$ of \ref{['cp']}.
• Figure 5: For $\gamma=3$, the energy $\mathcal{F}_i(p_t)-\mathcal{F}_i(p_0)$ of 500 trajectories.

Theorems & Definitions (28)

• Remark 2.2
• Proposition 2.3
• proof
• Lemma 2.4
• Definition 3.1: Generalized entropy and generalized relative entropy
• Remark 3.2
• Definition 3.3: Modified Fisher information and relative modified Fisher information
• Theorem 3.4
• Corollary 3.5
• Corollary 3.6