Existence and uniqueness theorems for solutions of McKean–Vlasov stochastic equations

New weak and strong existence and weak and strong uniqueness results for multi-dimensional stochastic McKean--Vlasov equations are established under relaxed regularity conditions. Weak existence is a variation of Krylov's weak existence for Ito's SDEs under the nondegeneracy of diffusion and no more than a linear growth in the state variable; this part is designed to fill in the existing gap, as earlier such results for McKean-Vlasov equations were not written. Weak and strong uniqueness is established under the restricted assumption of diffusion depending only on time and the state variable, yet without any regularity of the drift in the state variable and also under a linear growth condition on the drift; this part is based on the analysis of the total variation metric.

[1]  F. Smithies.  Linear Operators , 2019, Nature.

[2]  M. Kac.  Foundations of Kinetic Theory , 1956 .

[3]  C. Chou.  The Vlasov equations , 1965 .

[4]  A. Skorokhod,et al.  Studies in the theory of random processes , 1966 .

[5]  H. McKean,et al.  A CLASS OF MARKOV PROCESSES ASSOCIATED WITH NONLINEAR PARABOLIC EQUATIONS , 1966, Proceedings of the National Academy of Sciences of the United States of America.

[6]  M. Hp.  A class of markov processes associated with nonlinear parabolic equations. , 1966 .

[7]  R. K. Getoor.  Review: A. V. Skorokhod, Studies in the Theory of Random Processes , 1967 .

[8]  A. Vlasov,et al.  The vibrational properties of an electron gas , 1967, Uspekhi Fizicheskih Nauk.

[9]  N. V. Krylov,et al.  On Itô’s Stochastic Integral Equations , 1969 .

[10]  N. Krylov.  ON THE SELECTION OF A MARKOV PROCESS FROM A SYSTEM OF PROCESSES AND THE CONSTRUCTION OF QUASI-DIFFUSION PROCESSES , 1973 .

[11]  N. V. Krylov.  Addendum: On Itô’s Stochastic Integral Equations , 1973 .

[12]  On explicit formulas for solutions of stochastic equations , 1976 .

[13]  A. Veretennikov.  On the Strong Solutions of Stochastic Differential Equations , 1980 .

[14]  R. Khasminskii.  Stochastic Stability of Differential Equations , 1980 .

[15]  池田 信行,et al.  Stochastic differential equations and diffusion processes , 1981 .

[16]  A. Veretennikov.  ON STRONG SOLUTIONS AND EXPLICIT FORMULAS FOR SOLUTIONS OF STOCHASTIC INTEGRAL EQUATIONS , 1981 .

[17]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[18]  A. Veretennikov.  Parabolic equations and Itô's stochastic equations with coefficients discontinuous in the time variable , 1982 .

[19]  A. Yu. Veretennikov.  On the Criteria for Existence of a Strong Solution of a Stochastic Equation , 1983 .

[20]  Tadahisa Funaki,et al.  A certain class of diffusion processes associated with nonlinear parabolic equations , 1984 .

[21]  B. Øksendal.  Stochastic Differential Equations , 1985 .

[22]  H. Osada.  Diffusion processes with generators of generalized divergence form , 1987 .

[23]  A. Sznitman.  Topics in propagation of chaos , 1991 .

[24]  École d'été de probabilités de Saint-Flour,et al.  Ecole d'été de probabilités de Saint-Flour XIX, 1989 , 1991 .

[25]  O. A. Ladyzhenskai︠a︡,et al.  Linear and Quasi-linear Equations of Parabolic Type , 1995 .

[26]  R. Bass.  Diffusions and Elliptic Operators , 1997 .

[27]  Mireille Bossy,et al.  A stochastic particle method for the McKean-Vlasov and the Burgers equation , 1997, Math. Comput..

[28]  B. Jourdain.  Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized burgers' equations , 1997 .

[29]  D. Talay,et al.  Nonlinear self-stabilizing processes – I Existence, invariant probability, propagation of chaos , 1998 .

[30]  S. Shreve,et al.  Stochastic differential equations , 1955, Mathematical Proceedings of the Cambridge Philosophical Society.

[31]  Benjamin Jourdain,et al.  Propagation of chaos and fluctuations for a moderate model with smooth initial data , 1998 .

[32]  N. Krylov,et al.  Introduction to the Theory of Random Processes , 2002 .

[33]  V. Borkar.  Controlled diffusion processes , 2005, math/0511077.

[34]  D. Crisan,et al.  Approximate McKean–Vlasov representations for a class of SPDEs , 2005, math/0510668.

[35]  A. Veretennikov,et al.  On Ergodic Measures for McKean-Vlasov Stochastic Equations , 2006 .

[36]  Strong Solutions of Stochastic Differential Equations , 2006 .

[37]  A. Davie.  Uniqueness of solutions of stochastic differential equations , 2007, 0709.4147.

[38]  Aarnout Brombacher,et al.  Probability... , 2009, Qual. Reliab. Eng. Int..

[39]  On stochastic averaging and mixing , 2010 .

[40]  M. Manhart,et al.  Markov Processes , 2018, Introduction to Stochastic Processes and Simulation.

[41]  Stochastic Equations for Complex Systems , 2012 .

[42]  Thomas G. Kurtz,et al.  Weak and strong solutions of general stochastic models , 2013, 1305.6747.