We prove that the solution of the Vlasov-Fokker-Planck equation converges to the unique stationary solution with same mass as time tends to infinity. The same result holds in the repulsive coulombic case for the Vlasov-Poisson-Fokker-Planck system; the newtonian attractive case is also studied. We establish positive and negative answers to the question of existence of a stationary solution for ...

This is the third paper in a series concerning the study of steady states of a Fokker-Planck equation in a general domain in R with Lp̄loc drift term and W 1,p̄ loc diffusion term for any p̄ > n. In this paper, we give some existence results of stationary measures of the Fokker-Planck equation under Lyapunov conditions which allow the degeneracy of diffusion.

We introduce a stochastic particle system that corresponds to the Fokker–Planck equation with decay in the many-particle limit, and study its large deviations. We show that the large-deviation rate functional corresponds to an energy-dissipation functional in a Mosco-convergence sense. Moreover, we prove that the resulting functional, which involves entropic terms and the Wasserstein metric, is...

The fractional Fokker-Planck equation is often used to characterize anomalous diffusion. In this paper, a fully discrete approximation for the nonlinear spatial fractional Fokker-Planck equation is given, where the discontinuous Galerkin finite element approach is utilized in time domain and the Galerkin finite element approach is utilized in spatial domain. The priori error estimate is derived...

A Langevin equation with variable drift and diffusion coefficients separable in time and space and its corresponding Fokker-Planck equation in the Stratonovich approach are considered. From this Fokker-Planck equation a class of exact solutions with the same spatial drift and diffusion coefficients is obtained. Furthermore, some details of this system are analyzed by using the spatial diffusion...

We describe quantum-field-theoretical (QFT) techniques for mapping quantum problems onto c-number stochastic problems. This approach yields results which are identical to phase-space techniques [C.W. Gardiner, Quantum Noise (1991)] when the latter result in a Fokker-Planck equation for a corresponding pseudo-probability distribution. If phase-space techniques do not result in a Fokker-Planck eq...

We consider the linear Fokker-Planck equation in a confining potential in space dimension d ≥ 3. Using spectral methods, we prove bounds on the derivatives of the solution for short and long time, and give some applications. Résumé: On considère l’équation de Fokker-Planck avec un potentiel confinant en dimension d ≥ 3. Avec des méthodes spectrales on donne des bornes sur les dérivées de la sol...

Supplementary Material A. Background on Fokker-Planck Equation The Fokker-Planck equation (FPE) associated with a given stochastic differential equation (SDE) describes the time evolution of the distribution on the random variables under the specified stochastic dynamics. For example, consider the SDE: dz = g(z)dt+N (0, 2D(z)dt), (16) where z ∈ R, g(z) ∈ R, D(z) ∈ Rn×n. The distribution of z go...

The enhancement of the perpendicular temperature inside the resonant region, observed in numerical studies of the two-dimensional Fokker-Planck equation, combined with unidirectional RF quasilinear diffusion, is modeled on the basis of the collisional relaxation equations. Strong RF diffusion is assumed and relativistic effects are taken into account. The resulting enhanced perpendicular temper...

We derive, from first principle, the Fokker-Planck equation describing the non-perturbative real-time evolution of a gauge field at high temperatures on the spatial scale (g 2 T) −1 and the time scale (g 4 T) −1 , where g is the gauge coupling and T is the temperature. The knowledge of the effective dynamics on such spatial and time scales is crucial for the understanding and quantitative calcu...