Existence and stability for Fokker–Planck equations with log-concave reference measure

Abstract

We study Markov processes associated with stochastic differential equations, whose nonlinearities are gradients of convex functionals. We prove a general result of existence of such Markov processes and a priori estimates on the transition probabilities. The main result is the following stability property: if the associated invariant measures converge weakly, then the Markov processes converge in law. The proofs are based on the interpretation of a Fokker-Planck equation as the steepest descent flow of the relative Entropy functional in the space of probability measures, endowed with the Wasserstein distance. Applications include stochastic partial differential equations and convergence of equilibrium fluctuations for a class of random interfaces.

Variational approach to coarse-graining of generalized gradient flows

“…This shows how the structure of the relative Fisher Information is to some extent 'built-in' in this system. Relation with other variational formulations Our variational formulation (2) to 'passing to a limit' is closely related to other variational formulations in the literature, notably the - * formulation and the method in [7,64]. In the - * formulation, a gradient flow of the energy E ε : Z → R with respect to the dissipation * ε is defined to be a curve…”

“…The method introduced in [7,64] is slightly different. Therein 'passing to a limit' in the evolution equation is executed by studying (Gamma-)limits of the functionals that appear in the approximating discrete minimizing-movement schemes.…”

Passing to the limit in a Wasserstein gradient flow: from diffusion to reaction

“…This limit energy E free 0 is the Gamma-limit of E free ε [4]. However, the limit functional J 0 (·; a, b), does not have the same duality structure as (1.11), and we discuss this next.…”

“…Therefore one cannot canonically separate these two contributions in the structure of J 0 . This fact has another interesting consequence: in the present setting it is not possible to investigate separately the limit behaviour of the distance and of the functional using -convergence tools (as in the well-behaved gradient flows considered by [4,35,37,39]). Conversely, the geometry perturbed by the sublevels and by the slopes of the varying entropy functionals E free ε induces a new kind of evolution in the limit, which can solely be captured by considering the asymptotics of the whole space-time functionals J ε .…”

“…[4,28,30,39,38,40]). The inherent convexity and lower-semicontinuity properties of this type of formulation provide handles for such limit passages that are similar to the well-known results for elliptic systems-as we show below.…”