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

Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance

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.

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

[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.

[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.

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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.

[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.

[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.

[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.

[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.

[9] I. Csiszár and J. Körne, Information Theory: Coding Theorems for Discrete Memory-less Systems, Academic Press, New York, 1981. 
[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.

[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.

[12]

X. HuangM. 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.

[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.

[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.

[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.

[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.

[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.

[18]

Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, preprint, arXiv: 1603.02212.

[19]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964.

[20]

M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, preprint, arXiv: 1809.02216.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

A. K. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift, Math. Sb., 93 (1974), 129-149. 

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[9] I. Csiszár and J. Körne, Information Theory: Coding Theorems for Discrete Memory-less Systems, Academic Press, New York, 1981. 
[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.

[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.

[12]

X. HuangM. 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.

[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.

[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.

[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.

[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.

[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.

[18]

Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, preprint, arXiv: 1603.02212.

[19]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964.

[20]

M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, preprint, arXiv: 1809.02216.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

A. K. Zvonkin, A transformation of the phase space of a diffusion process that removes the drift, Math. Sb., 93 (1974), 129-149. 

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