Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance

Xing Huang; Feng-Yu Wang

doi:10.3934/dcds.2020336

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April 2021, 41(4): 1667-1679. doi: 10.3934/dcds.2020336

Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance

Xing Huang ^1, and Feng-Yu Wang ^2,,

1.	Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
2.	Center for Applied Mathematics, Tianjin University, Tianjin 300072, China, Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, United Kingdom

^* Corresponding author: Feng-Yu Wang

Received March 2020 Revised July 2020 Published April 2021 Early access October 2020

Fund Project: Supported in part by NNSFC (11771326, 11831014, 11801406, 11921001)

Existence and uniqueness are proved for McKean-Vlasov type distribution dependent SDEs with singular drifts satisfying an integrability condition in space variable and the Lipschitz condition in distribution variable with respect to $ {\mathbb W}_0 $ or $ {\mathbb W}_0+{\mathbb W}_\theta $ for some $ \theta\ge 1 $, where $ {\mathbb W}_0 $ is the total variation distance and $ {\mathbb W}_\theta $ is the $ L^\theta $-Wasserstein distance. This improves some existing results (see for instance [13]) derived for drifts continuous in the distribution variable with respect to the Wasserstein distance.

Keywords: Distribution dependent SDEs, total variation distance, Wasserstein distance, Krylov's estimate, Zvonkin's transform.

Mathematics Subject Classification: Primary: 60H10; Secondary: 60G44.

Citation: Xing Huang, Feng-Yu Wang. Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1667-1679. doi: 10.3934/dcds.2020336

References:

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[6]	P. E. Chaudru de Raynal, Strong well-posedness of McKean-Vlasov stochastic differential equation with Hölder drift, Stoch. Process Appl., 130 (2020), 79-107. doi: 10.1016/j.spa.2019.01.006. Google Scholar
[7]	G. Crippa and C. De Lellis, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., 616 (2008), 15-46. doi: 10.1515/CRELLE.2008.016. Google Scholar
[8]	L. Campi and M. Fischer, $N$-player games and mean-field games with absorption, Ann. Appl. Probab., 28 (2016), 2188-2242. doi: 10.1214/17-AAP1354. Google Scholar
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[12]	X. Huang, M. Röckner and F.-Y. Wang, Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs, Discrete Contin. Dyn. Syst., 39 (2019), 3017-3035. doi: 10.3934/dcds.2019125. Google Scholar
[13]	X. Huang and F.-Y. Wang, Distribution dependent SDEs with singular coefficients, Stoch. Process Appl., 129 (2019), 4747-4770. doi: 10.1016/j.spa.2018.12.012. Google Scholar
[14]	B. Jourdain, Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations, ESAIM Probab. Statist., 1 (1997), 339-355. doi: 10.1051/ps:1997113. Google Scholar
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[17]	D. Lacker, On a strong form of propagation of chaos for McKean-Vlasov equations, Electron. Commun. Probab., 23 (2018), 11 pp. doi: 10.1214/18-ECP150. Google Scholar
[18]	Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, preprint, arXiv: 1603.02212. Google Scholar
[19]	M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964. Google Scholar
[20]	M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, preprint, arXiv: 1809.02216. Google Scholar
[21]	A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Sain-Flour XIX-1989, Lecture Notes in Mathematics, 1464, Springer, Berlin, 1991,165–251. doi: 10.1007/BFb0085169. Google Scholar
[22]	F.-Y. Wang, Gradient estimate and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift, J. Differential Equations, 260 (2016), 2792-2829. doi: 10.1016/j.jde.2015.10.020. Google Scholar
[23]	F.-Y. Wang, Distribution-dependent SDEs for Landau type equations, Stoch. Proc. Appl., 128 (2018), 595-621. doi: 10.1016/j.spa.2017.05.006. Google Scholar
[24]	L. Xie and X. Zhang, Ergodicity of stochastic differential equations with jumps and singular coefficients, Ann. Inst. H. Poincaré Probab. Statist., 56 (2020), 175-229. doi: 10.1214/19-AIHP959. Google Scholar
[25]	C. G. Yuan and S.-Q. Zhang, A study on Zvonkin's transformation for stochastic differential equations with singular drift and related applications, preprint, arXiv: 1910.05903. Google Scholar
[26]	P. Xia, L. Xie, X. Zhang and G. Zhao, $L^q$($L^p$)-theory of stochastic differential equations, doi: 10.1016/j.spa.2020.03.004. Google Scholar
[27]	X. Zhang, Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients, Electron. J. Probab., 16 (2011), 1096-1116. doi: 10.1214/EJP.v16-887. Google Scholar
[28]	A. K. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift, Math. Sb., 93 (1974), 129-149. Google Scholar

show all references

References:

[1]	V. Barbu and M. Röckner, Probabilistic representation for solutions to non-linear Fokker-Planck equations, SIAM J. Math. Anal., 50 (2018), 4246-4260. doi: 10.1137/17M1162780. Google Scholar
[2]	V. Barbu and M. Röckner, From non-linear Fokker-Planck equations to solutions of distribution dependent SDE, Ann. Probab., 48 (2020), 1902-1920. doi: 10.1214/19-AOP1410. Google Scholar
[3]	M. Bauer and T. M-Brandis, Existence and regularity of solutions to multi-dimensional mean-field stochastic differential equations with irregular drift, arXiv: 1912.05932. Google Scholar
[4]	M. Bauer, T. M-Brandis and F. Proske, Strong solutions of mean-field stochastic differential equations with irregular drift, Electron. J. Probab., 23 (2018), 35 pp. doi: 10.1214/18-EJP259. Google Scholar
[5]	K. Carrapatoso, Exponential convergence to equilibrium for the homogeneous Landau equation with hard potentials, Bull. Sci. Math., 139 (2015), 777-805. doi: 10.1016/j.bulsci.2014.12.002. Google Scholar
[6]	P. E. Chaudru de Raynal, Strong well-posedness of McKean-Vlasov stochastic differential equation with Hölder drift, Stoch. Process Appl., 130 (2020), 79-107. doi: 10.1016/j.spa.2019.01.006. Google Scholar
[7]	G. Crippa and C. De Lellis, Estimates and regularity results for the DiPerna-Lions flow, J. Reine Angew. Math., 616 (2008), 15-46. doi: 10.1515/CRELLE.2008.016. Google Scholar
[8]	L. Campi and M. Fischer, $N$-player games and mean-field games with absorption, Ann. Appl. Probab., 28 (2016), 2188-2242. doi: 10.1214/17-AAP1354. Google Scholar
[9]	I. Csiszár and J. Körne, Information Theory: Coding Theorems for Discrete Memory-less Systems, Academic Press, New York, 1981. Google Scholar
[10]	L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials, Part I: existence, uniqueness and smothness, Comm. Part. Diff. Equat., 25 (2000), 179-259. doi: 10.1080/03605300008821512. Google Scholar
[11]	L. Desvillettes and C. Villani, On the spatially homogeneous Landau equation for hard potentials, part II: H-theorem and applications, Comm. Part. Diff. Equat., 25 (2000), 261-298. doi: 10.1080/03605300008821513. Google Scholar
[12]	X. Huang, M. Röckner and F.-Y. Wang, Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs, Discrete Contin. Dyn. Syst., 39 (2019), 3017-3035. doi: 10.3934/dcds.2019125. Google Scholar
[13]	X. Huang and F.-Y. Wang, Distribution dependent SDEs with singular coefficients, Stoch. Process Appl., 129 (2019), 4747-4770. doi: 10.1016/j.spa.2018.12.012. Google Scholar
[14]	B. Jourdain, Diffusions with a nonlinear irregular drift coefficient and probabilistic interpretation of generalized Burgers' equations, ESAIM Probab. Statist., 1 (1997), 339-355. doi: 10.1051/ps:1997113. Google Scholar
[15]	B. Jourdain and S. Méléard, Propagation of chaos and fluctuations for a moderate model with smooth initial data, Ann. Inst. H. Poincaré Probab. Statist., 34 (1998), 727-766. doi: 10.1016/S0246-0203(99)80002-8. Google Scholar
[16]	N. V. Krylov and M. Röckner, Strong solutions of stochastic equations with singular time dependent drift, Probab. Theory Related Fields, 131 (2005), 154-196. doi: 10.1007/s00440-004-0361-z. Google Scholar
[17]	D. Lacker, On a strong form of propagation of chaos for McKean-Vlasov equations, Electron. Commun. Probab., 23 (2018), 11 pp. doi: 10.1214/18-ECP150. Google Scholar
[18]	Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, preprint, arXiv: 1603.02212. Google Scholar
[19]	M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964. Google Scholar
[20]	M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, preprint, arXiv: 1809.02216. Google Scholar
[21]	A.-S. Sznitman, Topics in propagation of chaos, in École d'Été de Probabilités de Sain-Flour XIX-1989, Lecture Notes in Mathematics, 1464, Springer, Berlin, 1991,165–251. doi: 10.1007/BFb0085169. Google Scholar
[22]	F.-Y. Wang, Gradient estimate and applications for SDEs in Hilbert space with multiplicative noise and Dini continuous drift, J. Differential Equations, 260 (2016), 2792-2829. doi: 10.1016/j.jde.2015.10.020. Google Scholar
[23]	F.-Y. Wang, Distribution-dependent SDEs for Landau type equations, Stoch. Proc. Appl., 128 (2018), 595-621. doi: 10.1016/j.spa.2017.05.006. Google Scholar
[24]	L. Xie and X. Zhang, Ergodicity of stochastic differential equations with jumps and singular coefficients, Ann. Inst. H. Poincaré Probab. Statist., 56 (2020), 175-229. doi: 10.1214/19-AIHP959. Google Scholar
[25]	C. G. Yuan and S.-Q. Zhang, A study on Zvonkin's transformation for stochastic differential equations with singular drift and related applications, preprint, arXiv: 1910.05903. Google Scholar
[26]	P. Xia, L. Xie, X. Zhang and G. Zhao, $L^q$($L^p$)-theory of stochastic differential equations, doi: 10.1016/j.spa.2020.03.004. Google Scholar
[27]	X. Zhang, Stochastic homeomorphism flows of SDEs with singular drifts and Sobolev diffusion coefficients, Electron. J. Probab., 16 (2011), 1096-1116. doi: 10.1214/EJP.v16-887. Google Scholar
[28]	A. K. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift, Math. Sb., 93 (1974), 129-149. Google Scholar

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