American Institute of Mathematical Sciences

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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  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 - A, doi: 10.3934/dcds.2020336
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

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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

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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

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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

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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

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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

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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
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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

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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

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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

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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

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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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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

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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

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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

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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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