Convergence rates of Wasserstein gradient flows for nonlinear Fokker-Planck equations with mobility and related inequalities

Abstract: For nonlinear Fokker-Planck equations with mobility, the Wasserstein gradient flow structure is described by the generalized relative entropy as the energy functional and the modified Wasserstein metric $W_h$ as the associated metric structure. This work investigates the nonlinear effects induced by mobility and establishes the corresponding inequalities. For nonlinear diffusion, we establish a logarithmic Sobolev inequality, which yields the convergence rate of the free energy functional and the Talagrand inequality. By further exploiting the relationship between the weighted homogeneous Sobolev norm and the $W_h$ metric, we derive an HWI inequality relating the relative entropy, the $W_h$ metric, and the Fisher information. In the case of mobility dependent drift and linear diffusion, the convergence rate in the $W_h$ metric is also obtained by applying the Girsanov theorem.

• The paper establishes exponential convergence of free energy and transport distances for NFPEs using a generalized Wasserstein gradient flow framework.
• It introduces novel inequalities—logarithmic Sobolev, Talagrand, and HWI—tailored to handle nonlinear mobility and non-displacement convexity.
• The work integrates PDE analysis with stochastic techniques like the Girsanov transform to rigorously control convergence under mobility-dependent dynamics.

Convergence Rates of Nonlinear Fokker-Planck Equations: Wasserstein Gradient Flows with Mobility

Problem Formulation and Gradient Structure

The paper studies nonlinear Fokker-Planck equations (NFPEs) with mobility: $$\partial_t p_t = \mathrm{div}(\nabla \Phi\, h(p_t)) + \Delta f(p_t),$$ where $h(r) = rb(r)$ encodes a nonlinear "mobility" and $f$ generates the nonlinear diffusion. The equation is coupled with the McKean-Vlasov SDE exhibiting law-dependent drift and nonlinear diffusion. Mobility terms are critical for modeling crowding effects and yield a generalized Wasserstein gradient flow structure: $$\partial_t p_t = -\mathrm{grad}_{W_h} \mathcal{F}(p_t) = \mathrm{div} \big( h(p_t) \nabla ( \delta \mathcal{F}(p_t) / \delta p ) \big).$$ The associated free energy is

$\mathcal{F}(p) = \int \eta(p) + \Phi p\, dx,$

where $\eta$ is the generalized entropy integrand. The natural metric is a weighted Wasserstein-type metric $W_h$, interpolating between classical cases and incorporating the mobility dependence.

A crucial technical aspect is dealing with functionals (such as the potential energy and relative entropy) that are not displacement convex due to mobility, impeding classical convergence analysis in Wasserstein spaces.

Main Results: Inequalities and Exponential Convergence

The authors develop a robust framework for analyzing convergence by restricting to a suitable admissible set of probability measures, $\Lambda^g$, ensuring strong uniform control on solutions: $$\Lambda^g := \{ P = p\, dx \mid \| g(p) - g(p_\infty) \|_{L^\infty} \leq C \},$$ with $g$ the entropy generator and $h(r) = rb(r)$0 the stationary solution. This avoids the necessity of (generalized) displacement convexity.

Logarithmic Sobolev Inequality and Convergence Rates

A key technical contribution is the establishment of a logarithmic Sobolev inequality of the type: $h(r) = rb(r)$1 where $h(r) = rb(r)$2 is the generalized relative entropy and $h(r) = rb(r)$3 is a modified Fisher information incorporating the mobility structure. This generalizes classical LSI, exactly quantifying nonlinear effects induced by $h(r) = rb(r)$4.

The entropy dissipation identity,

$h(r) = rb(r)$5

combined with LSI, leads via Gronwall's lemma to exponential convergence of the free energy: $h(r) = rb(r)$6 This provides explicit rates, highlighting the robust decay of entropy and hence convergence to equilibrium, even beyond displacement convexity settings.

Talagrand and HWI Inequalities

The work establishes a Talagrand inequality for the modified metric: $h(r) = rb(r)$7 with $h(r) = rb(r)$8 explicit and dependent on mobility. This links entropy to transport distance in this generalized setting.

Furthermore, by leveraging connections between weighted Sobolev norms, $h(r) = rb(r)$9, and the underlying dynamics, the authors prove a generalized HWI inequality: $f$0 encompassing the mobility term. The construction uses admissible sets and avoids convexity obstructions by delicate analysis, including a comparison with standard $f$1 distances and properties of log-concave stationary measures.

Mobility-Dependent Drift, Linear Diffusion, and Girsanov Techniques

In the case where the drift is mobility dependent but the diffusion is linear, the analysis exploits stochastic calculus techniques. Applying the Girsanov theorem, the authors are able to construct explicit exponential convergence rates in the $f$2 metric under suitable regularity and growth conditions on the potential $f$3. The existence of a log-Harnack inequality is established, and sharp upper bounds for the convergence of $f$4 are obtained, yielding exponential convergence in the transport metric.

Technical Innovations

• The approach circumvents the lack of displacement convexity by rigorously restricting trajectory space to sets where direct estimates are tractable. This is shown necessary: counterexamples demonstrate that balls in the $f$5 metric are not sufficient to preserve the invariance and structure required.
• All constants in the main inequalities (LSI, Talagrand, HWI) are made explicit in terms of the nonlinearities, admissible set parameters, and regularity of the problem data.
• The method skillfully integrates PDE estimates, Wasserstein geometry, and stochastic analysis (via the Girsanov transform) to derive convergence rates for a broad class of nonlinear PDEs.

Implications and Future Directions

The results provide a rigorous and quantitative framework for analyzing nonlinear Fokker-Planck equations with degenerate mobility at the level of both free energy and transport metrics. This is directly relevant to interacting particle systems and mean-field models with crowding and exclusion processes, as well as variational inference for nonlinear partial differential equations.

The implications are:

• The explicit convergence rates in the generalized Wasserstein metric advance understanding of equilibration in non-displacement convex settings, which have been out of the reach of classical optimal transport techniques.
• The inequalities and technical tools developed can likely be adapted to other classes of nonlinear PDEs, especially where mobility is required by modeling constraints (e.g., degenerate diffusions, aggregation-diffusion equations).
• The stochastic coupling and Girsanov-based analysis may be extended to quantify rates for higher moments and pathwise quantitative propagation of chaos in mean-field limits of interacting SDEs.

Open problems include weakening invariance requirements on the admissible set, extending to multiple stationary states, and deriving analogous results in bounded domains or non-compact settings. Further, applications to nonlinear nonlocal drift-diffusion systems and problems in high-dimensional statistical physics may benefit directly from these techniques.

Conclusion

This paper systematically analyzes the convergence of Wasserstein gradient flows of NFPEs with mobility, establishing sharp inequalities and exponential convergence for the nonlinear entropy and transport distance, even in the absence of displacement convexity. The results rely on careful analysis of admissible classes, the development of generalized metric and entropy dissipation tools, and stochastic coupling methods to handle mobility-induced nonlinearities. The approach forms a rigorous foundation for further study of nonlinear evolution equations in transport and diffusion theory, with potential for significant extension in both theory and applications.