Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations

@article{Bolley2012ConvergenceTE,
  title={Convergence to equilibrium in Wasserstein distance for Fokker-Planck equations},
  author={Franccois Bolley and Ivan Gentil and Arnaud Guillin},
  journal={Journal of Functional Analysis},
  year={2012},
  volume={263},
  pages={2430-2457}
}

Convergence to equilibrium for the kinetic Fokker-Planck equation on the torus

We study convergence to equilibrium for the kinetic Fokker-Planck equation on the torus. Solving the stochastic differential equation, we show exponential convergence in the

Convergence to Equilibrium in Wasserstein Distance for Damped Euler Equations with Interaction Forces

We develop tools to construct Lyapunov functionals on the space of probability measures in order to investigate the convergence to global equilibrium of a damped Euler system under the influence of

Contraction in the Wasserstein metric for the kinetic Fokker-Planck equation on the torus

We study contraction for the kinetic Fokker-Planck operator on the torus. Solving the stochastic differential equation, we show contraction and therefore exponential convergence in the

The Kantorovich and variation distances between invariant measures of diffusions and nonlinear stationary Fokker-Planck-Kolmogorov equations

We obtain upper bounds for the total variation distance and the quadratic Kantorovich distance between stationary distributions of two diffusion processes with different drifts. More generally, our

An optimal transport approach of hypocoercivity for the 1d kinetic Fokker-Plank equation

A quadratic optimal transport metric on the set of probability measure over $\R^2$ is introduced. The quadratic cost is given by the euclidean norm on $\R^2$ associated to some well chosen symmetric

Stein's method for steady-state diffusion approximation in Wasserstein distance

A general steady-state diffusion approximation result is provided which bounds the Wasserstein distance between the reversible measure μ of a diffusion process and the invariant measure ν of a Markov chain, providing a quantitative answer to a problem of interest to the machine learning community.

Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schrödinger equation

We consider the dispersive logarithmic Schrodinger equation in a semi-classical scaling. We extend the results about the large time behaviour of the solution (dispersion faster than usual with an

Wasserstein stability estimates for covariance-preconditioned Fokker–Planck equations

We study the convergence to equilibrium of the mean field PDE associated with the derivative-free methodologies for solving inverse problems that are presented by Garbuno-Inigo et al (2020 SIAM J.

Wasserstein Contraction of Stochastic Nonlinear Systems

We suggest that the tools of contraction analysis for deterministic systems can be applied towards studying the convergence behavior of stochastic dynamical systems in the Wasserstein metric. In
...

References

SHOWING 1-10 OF 31 REFERENCES

Trend to equilibrium and particle approximation for a weakly selfconsistent Vlasov-Fokker-Planck equation

We consider a Vlasov-Fokker-Planck equation governing the evolution of the density of interacting and diffusive matter in the space of positions and velocities. We use a probabilistic interpretation

Large-time behavior of non-symmetric Fokker-Planck type equations

Large time asymptotics of the solutions to non-symmetric Fokker- Planck type equations are studied by extending the entropy method to this case. We present a modified Bakry-Emery criterion that

Contractions in the 2-Wasserstein Length Space and Thermalization of Granular Media

An algebraic decay rate is derived which bounds the time required for velocities to equilibrate in a spatially homogeneous flow-through model representing the continuum limit of a gas of particles

Strict contractivity of the 2-wasserstein distance for the porous medium equation by mass-centering

We show that the Euclidean Wasserstein distance between two compactly supported solutions of the one-dimensional porous medium equation having the same center of mass decays to zero for large times.

Uniform Convergence to Equilibrium for Granular Media

We study the long time asymptotics of a nonlinear, nonlocal equation used in the modelling of granular media. We prove a uniform exponential convergence to equilibrium for degenerately convex and

Transport inequalities, gradient estimates, entropy and Ricci curvature

We present various characterizations of uniform lower bounds for the Ricci curvature of a smooth Riemannian manifold M in terms of convexity properties of the entropy (considered as a function on the

Refined long-time asymptotics for some polymeric fluid flow models

We consider a polymeric fluid model, consisting of the incompressible NavierStokes equations coupled to a non-symmetric Fokker-Planck equation. First, steady states and exponential convergence to

Some Applications of Mass Transport to Gaussian-Type Inequalities

This map is used in the setting of Gaussian-like measures to derive an inequality linking entropy with mass displacement by a straightforward argument and logarithmic Sobolev and transport inequalities are recovered.