Fokker–Planck Equations

Abstract

In this chapter, we review the Bakry–Emery approach from the PDE viewpoint (Sect. 2.1) and the original stochastic viewpoint (Sect. 2.3) and detail some known relations to convex Sobolev inequalities (Sect. 2.2). Our focus is the PDE viewpoint addressed by (Toscani, G, Entropy production and the rate of convergence to equilibrium for the Fokker-Planck equation, Quart. Appl. Math, 57, 521–541, (1999) [43]), and we follow partially the presentation of Matthes, D, Entropy Methods and Related Functional Inequalities, Lecture Notes, Pavia, Italy, (2007) http://www-m8.ma.tum.de/personen/matthes/papers/lecpavia.pdf [34]. The original Bakry–Emery method in (Bakry, D, Emery, M, Diffusions hypercontractives. Seminaire de probabilites XIX, 1983/84, Lecture Notes in Mathmatics, vol. 1123, pp. 177–206, Springer, Berlin (1985) [7] has been elaborated by many authors, and we select some of its extensions, including intermediate asymptotics by Carrillo and Toscani, Asymptotic \(L^1\)-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49, 113–142, (2000) [13] (Sect. 2.4) and more general Fokker–Planck equations, investigated, e.g., by Carrillo et al., Asymptotic \(L^1\)-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49, 113–142, (2000); Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133, 1–82, (2001) [13, 14] and Arnold et al., Large-time behavior of non-symmetric Fokker-Planck type equations, Commun. Stoch. Anal., 2, 153–175, (2008); Sharp entropy decay for hypocoercive and non-symmetric Fokker–Planck equations with linear drift, Preprint (2014). arXiv:1409.5425 [2, 5] (Sects. 2.5–2.6). Because of limited space, we ignore many important developments and deep connections to, e.g., optimal transport and Riemannian geometry, and we just refer to Villani, C.: Optimal Transport Old and New. Springer, Berlin (2009) [46] and the numerous references therein for more information.