Distribution dependent stochastic differential equations

Abstract

Due to their intrinsic link with nonlinear Fokker-Planck equations and many other applications, distribution dependent stochastic differential equations (DDSDEs) have been intensively investigated. In this paper, we summarize some recent progresses in the study of DDSDEs, which include the correspondence of weak solutions and nonlinear Fokker-Planck equations, the well-posedness, regularity estimates, exponential ergodicity, long time large deviations, and comparison theorems.

Keywords

Distribution dependent stochastic differential equation (DDSDE) / nonlinear Fokker-Planck equation / Bismut formula / Wasserstein distance / gradient estimate

References

[1] Albeverio S, Kondratiev Y G, Röckner M. Differential geometry of Poisson spaces. C R Acad Sci Paris Sér I Math, 1996, 323: 1129–1134

[2] Bao J H, Ren P P, Wang F-Y. Bismut formulas for Lions derivative of McKean-Vlasov SDEs with memory. arXiv: 2004.14629

[3] Bao J H, Wang F-Y, Yuan C G. Hypercontractivity for functional stochastic partial differential equations. Electron Commun Probab, 2015, 20: 1–15

[4] Bao J H, Yuan C G. Comparison theorem for stochastic differential delay equations with jumps. Acta Appl Math, 2011, 116: 119–132

[5] Barbu V, Röckner M. Probabilistic representation for solutions to nonlinear Fokker- Planck equations. SIAM J Math Anal, 2018, 50: 4246–4260

[6] Barbu V, Röckner M. From nonlinear Fokker-Planck equations to solutions of distribution dependent SDE. Ann Probab, 2020, 48: 1902–1920

[7] Bobkov S G, Gentil I, Ledoux M. Hypercontractivity of Hamilton-Jacobi equations. J Math Pures Appl, 2001, 80: 669–696

[8] Bogachev V, Krylov N V, Röckner M. On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions. Comm Partial Differential Equations, 2001, 26: 2037–2080

[9] Cardaliaguet P. Notes on mean field games. Lions P-L Lectures at College de France. www.ceremade.dauphine.fr/∼cardaliaguet/MFG20130420.pdf

[10] Carrillo J A, McCann R J, Villani C. Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates. Rev Mat Iberoam, 2003, 19: 971–1018

[11] Chen M-F, Wang F-Y. On order-preservation and positive correlations for multi- dimensional diffusion processes. Probab Theory Related Fields, 1993, 95: 421–428

[12] de Raynal P-E C, Frikha N. Well-posedness for some non-linear diffusion processes and related pde on the Wasserstein space. arXiv: 1811.06904

[13] Dembo A, Zeitouni O. Large Deviations Techniques and Applications. 2nd ed. New York: Springer, 1998

[14] Donsker M D, Varadhan S R S. Asymptotic evaluation of certain Markov process expectations for large time, I. Comm Pure Appl Math, 1975, 28: 1–47

[15] Donsker M D, Varadhan S R S. Asymptotic evaluation of certain Markov process expectations for large time, II. Comm Pure Appl Math, 1975, 28: 279–301

[16] Donsker M D, Varadhan S R S. Asymptotic evaluation of certain Markov process expectations for large time, III. Comm Pure Appl Math, 1976, 29: 389–461

[17] Donsker M D, Varadhan S R S. Asymptotic evaluation of certain Markov process expectations for large time, IV. Comm Pure Appl Math, 1983, 36: 183–212

[18] Gal’cuk L, Davis M. A note on a comparison theorem for equations with different diffusions. Stochastics, 1982, 6: 147–149

[19] Guillin A, Liu W, Wu L M. Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems. Ann Appl Probab (to appear)

[20] Guillin A, Wang F-Y. Degenerate Fokker-Planck equations: Bismut formula, gradient estimate and Harnack inequality. J Differential Equations, 2012, 253: 20–40

[21] Hammersley W, Šiška D, Szpruch L. McKean-Vlasov SDE under measure dependent Lyapunov conditions. arXiv: 1802.03974

[22] Huang X, Liu C, Wang F-Y. Order preservation for path-distribution dependent SDEs. Commun Pure Appl Anal, 2018, 17: 2125–2133

[23] Huang X, Röckner M, Wang F-Y. Nonlinear Fokker-Planck equations for probability measures on path space and path-distribution dependent SDEs. Discrete Contin Dyn Syst, 2019, 39: 3017–3035

[24] Huang X, Wang F-Y. Order-preservation for multidimensional stochastic functional differential equations with jumps. J Evol Equ, 2014, 14: 445–460

[25] Huang X, Wang F-Y. Distribution dependent SDEs with singular coefficients. Stochastic Process Appl, 2019, 129: 4747–4770

[26] Huang X, Wang F-Y. McKean-Vlasov SDEs with drifts discontinuous under Wasserstein distance. Discrete Contin Dyn Syst, 2021, 41: 1667–1679

[27] Huang X, Wang F-Y. Derivative estimates on distributions of McKean-Vlasov SDEs. Electron J Probab (to appear)

[28] Huang X, Wang F-Y. Well-posedness for singular McKean-Vlasov stochastic differential equations. arXiv: 2012.05014

[29] Huang X, Yuan C. Comparison theorem for distribution dependent neutral SFDEs. J Evol Equ, 2020, https://doi.org/10.1007/s00028-020-00595-w

[30] Ikeda N, Watanabe S. A comparison theorem for solutions of stochastic differential equations and its applications. Osaka J Math, 1977, 14: 619–633

[31] Kac M. Foundations of kinetic theory. In: Proceedings of the Third Berkeley Symposium on Mathematical Statistics and Probability, 1954-1955, Vol III. Berkeley: Univ of California Press, 1956, 171–197

[32] Kac M. Probability and Related Topics in the Physical Sciences. New York: Interscience, 1959

[33] Kamae T, Krengel U, O’Brien G L. Stochastic inequalities on partially ordered spaces. Ann Probab, 1977, 5: 899–912

[34] Kurtz T. Weak and strong solutions of general stochastic models. Electron Commun Probab, 2014, 19: (16 pp)

[35] Lanconelli E, Polidoro S. On a class of hypoelliptic evolution operator. Rend Semin Mat Univ Politec Torino, 1994, 52: 29–63

[36] Mao X R. A note on comparison theorems for stochastic differential equations with respect to semimartingales. Stochastics, 1991, 37: 49–59

[37] McKean H P. A class of Markov processes associated with nonlinear parabolic equations. Proc Natl Acad Sci USA, 1966, 56: 1907–1911

[38] O’Brien G. A new comparison theorem for solution of stochastic differential equations. Stochastics, 1980, 3: 245–249

[39] Otto F, Villani C. Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality. J Funct Anal, 2000, 173: 361–400

[40] Ren P P, Tang H, Wang F-Y. Distribution-path dependent nonlinear SPDEs with application to stochastic transport type equations. arXiv: 2007.09188

[41] Ren P P, Wang F-Y. Bismut formula for Lions derivative of distribution dependent SDEs and applications. J Differential Equations, 2019, 267: 4745–4777

[42] Ren P P, Wang F-Y. Derivative formulas in measure on Riemannian manifolds. arXiv: 1908.03711

[43] Ren P P, Wang F-Y. Donsker-Varadhan large deviations for path-distribution dependent SPDEs. J Math Anal Appl (to appear)

[44] Ren P P, Wang F-Y. Exponential convergence in entropy and Wasserstein distance for McKean-Vlasov SDEs. Nonlinear Anal (to appear)

[45] Röckner M, Wang F-Y. Log-Harnack inequality for stochastic differential equations in Hilbert spaces and its consequences. Infin Dimens Anal Quantum Probab Relat Top, 2010, 13: 27–37

[46] Röckner M, Zhang X C. Well-posedness of distribution dependent SDEs with singular drifts. Bernoulli (to appear)

[47] Sznitman A S. Topics in propagations of chaos. In: Burkholder D L, Pardoux E, Sznitman A, eds. Ecole d’Eté de Probabilités de Saint-Flour XIX–1989. Lecture Notes in Math, Vol 1464. Berlin: Springer, 1991, 165–251

[48] Talagrand M. Transportation cost for Gaussian and other product measures. Geom Funct Anal, 1996, 6: 587–600

[49] Wang F-Y. Logarithmic Sobolev inequalities on noncompact Riemannian manifolds. Probab Theory Related Fields, 1997, 109: 417–424

[50] Wang F-Y. Probability distance inequalities on Riemannian manifolds and path spaces. J Funct Anal, 2004, 206: 167–190

[51] Wang F-Y. The stochastic order and critical phenomena for superprocesses. Infin Dimens Anal Quantum Probab Relat Top, 2006, 9: 107–128

[52] Wang F-Y. Harnack inequalities on manifolds with boundary and applications. J Math Pures Appl, 2010, 94: 304–321

[53] Wang F-Y. Harnack Inequalities and Applications for Stochastic Partial Differential Equations. Berlin: Springer, 2013

[54] Wang F-Y. Distribution dependent SDEs for Landau type equations. Stochastic Process Appl, 2018, 128: 595–621

[55] Wang F-Y, Zhang X C. Derivative formula and applications for degenerate diffusion semigroups. J Math Pures Appl, 2013, 99: 726–740

[56] Wang J-M. Stochastic comparison for Lévy-type processes. J Theoret Probab, 2013, 26: 997–1019

[57] Yamada T, Watanabe S. On the uniqueness of solutions of stochastic differential equations. J Math Kyoto U, 1971, 2: 155–167

[58] Yang Z, Mao X R, Yuan C G. Comparison theorem of one-dimensional stochastic hybrid systems. Systems Control Lett, 2008, 57: 56–63

[59] Zhang X C. Weak solutions of McKean-Vlasov SDEs with supercritical drifts. arXiv: 2010.15330

[60] Zhao G H. On distribution dependent SDEs with singular drifts. arXiv: 2003.04829

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