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

By Luigi Ambrosio, Giuseppe Savar'e, L. Zambotti. Probability theory and related fields, 2007

118 citations recorded by the source corpus.

We study Markov processes associated with stochastic differential equations, whose non-linearities 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.

Subfields: Applied Mathematics, Finance, Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/31bc6e372c12978e16f6bdd1242b217dd2eb010f

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Citations in the corpus

Distances between transition probabilities of diffusions and applications to nonlinear Fokker–Planck–Kolmogorov equations (2016, unknown): “, [1], [2], [16], [17], [18], [20], [21], [23], [30], and [33].”
Gradient flows of the entropy for jump processes (2012, background): “Recently, it has been shown that the gradient flow characterization provides a good framework to study stability properties of diffusion processes under changes of the driving potential or the underlying geometry [3,18].”
Optimal transport from Lebesgue to Poisson (2010, background): “racted lot of interest and has producedan enormous amount of deep results, striking applications and stimulating new developments, among others in PDEs (e.g., [3, 7, 23]), evolution semigroups (e.g., [4, 22, 24]) and geometry (e.g., [19, 21, 26, 27, 32]). Ajtai, Komlo´s and Tusn´ady as well as Talag”
CALCULUS, HEAT FLOW AND CURVATURE-DIMENSION BOUNDS IN METRIC MEASURE SPACES (2019, background): “At the level of CD spaces, we know from Sturm [2006b], Bacher and Sturm [2010], and Villani [2009] that essentially non-branching CD(0; N ), CD(K; 1) and CD (K; N ) spaces all have the tensorization property.”
Well posedness of Lagrangian flows and continuity equations in metric measure spaces (2014, background): “It is shown in [Ambrosio et al. 2009] that the Eγ is closable, extending previous results obtained under more restrictive assumptions on γ .”
Gradient flows of the entropy for finite Markov chains (2011, background): “By now a similar interpretation of the heat flow has been established in a wide variety of settings, including Riemannian manifolds [10], Hilbert spaces [2], Wiener spaces [11], Finsler spaces [19], Alexandrov spaces [13] and metric measure spaces [12, 25].”
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below (2011, unknown): “In connection with (ii), notice that the equivalence so far has been proved in Euclidean spaces by Jordan–Kinderleher–Otto, in the seminal paper [24], in Riemannian manifolds by Erbar and Villani [13, 41], in Hilbert spaces by [6], in Finsler spaces by Ohta–Sturm [32] and eventually in Alexandrov sp”
Wasserstein space over the Wiener space (2010, background): “Remark The Fokker–Planck equations and related topics on a Hilbert space were studied recently in [ 4 ].”