Mckean-Vlasov sdes with drifts discontinuous under wasserstein distance

Previous Article
Global stability in a multi-dimensional predator-prey system with prey-taxis
DCDS Home
This Issue
Next Article
Recurrence for measurable semigroup actions

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 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.
[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. 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.
[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. 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.
[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.

[1]	Xing Huang, Chang Liu, Feng-Yu Wang. Order preservation for path-distribution dependent SDEs. Communications on Pure and Applied Analysis, 2018, 17 (5) : 2125-2133. doi: 10.3934/cpaa.2018100
[2]	Qing Ma, Yanjun Wang. Distributionally robust chance constrained svm model with $\ell_2$-Wasserstein distance. Journal of Industrial and Management Optimization, 2021 doi: 10.3934/jimo.2021212
[3]	Yuyuan Ouyang, Yunmei Chen, Ying Wu. Total variation and wavelet regularization of orientation distribution functions in diffusion MRI. Inverse Problems and Imaging, 2013, 7 (2) : 565-583. doi: 10.3934/ipi.2013.7.565
[4]	Zhiwei Tian, Yanyan Shi, Meng Wang, Xiaolong Kong, Lei Li, Feng Fu. A wavelet frame constrained total generalized variation model for imaging conductivity distribution. Inverse Problems and Imaging, 2022, 16 (4) : 753-769. doi: 10.3934/ipi.2021074
[5]	Yulin Song. Density functions of distribution dependent SDEs driven by Lévy noises. Communications on Pure and Applied Analysis, 2021, 20 (6) : 2399-2419. doi: 10.3934/cpaa.2021087
[6]	Panpan Ren, Shen Wang. Moderate deviation principles for unbounded additive functionals of distribution dependent SDEs. Communications on Pure and Applied Analysis, 2021, 20 (9) : 3129-3142. doi: 10.3934/cpaa.2021099
[7]	Xing Huang, Yulin Song, Feng-Yu Wang. Bismut formula for intrinsic/Lions derivatives of distribution dependent SDEs with singular coefficients. Discrete and Continuous Dynamical Systems, 2022, 42 (9) : 4597-4614. doi: 10.3934/dcds.2022065
[8]	Claude Carlet. Expressing the minimum distance, weight distribution and covering radius of codes by means of the algebraic and numerical normal forms of their indicators. Advances in Mathematics of Communications, 2022 doi: 10.3934/amc.2022047
[9]	José M. Arrieta, Esperanza Santamaría. Estimates on the distance of inertial manifolds. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 3921-3944. doi: 10.3934/dcds.2014.34.3921
[10]	Liliana Trejo-Valencia, Edgardo Ugalde. Projective distance and $g$-measures. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3565-3579. doi: 10.3934/dcdsb.2015.20.3565
[11]	Xing Huang, Michael Röckner, Feng-Yu Wang. Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs. Discrete and Continuous Dynamical Systems, 2019, 39 (6) : 3017-3035. doi: 10.3934/dcds.2019125
[12]	Bao Wang, Alex Lin, Penghang Yin, Wei Zhu, Andrea L. Bertozzi, Stanley J. Osher. Adversarial defense via the data-dependent activation, total variation minimization, and adversarial training. Inverse Problems and Imaging, 2021, 15 (1) : 129-145. doi: 10.3934/ipi.2020046
[13]	Mario Roy. A new variation of Bowen's formula for graph directed Markov systems. Discrete and Continuous Dynamical Systems, 2012, 32 (7) : 2533-2551. doi: 10.3934/dcds.2012.32.2533
[14]	Hans-Otto Walther. On Poisson's state-dependent delay. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 365-379. doi: 10.3934/dcds.2013.33.365
[15]	Xiaoqun Zhang, Tony F. Chan. Wavelet inpainting by nonlocal total variation. Inverse Problems and Imaging, 2010, 4 (1) : 191-210. doi: 10.3934/ipi.2010.4.191
[16]	Konstantinos Drakakis, Roderick Gow, Scott Rickard. Common distance vectors between Costas arrays. Advances in Mathematics of Communications, 2009, 3 (1) : 35-52. doi: 10.3934/amc.2009.3.35
[17]	Chun-Xiang Guo, Guo Qiang, Jin Mao-Zhu, Zhihan Lv. Dynamic systems based on preference graph and distance. Discrete and Continuous Dynamical Systems - S, 2015, 8 (6) : 1139-1154. doi: 10.3934/dcdss.2015.8.1139
[18]	Yujuan Li, Guizhen Zhu. On the error distance of extended Reed-Solomon codes. Advances in Mathematics of Communications, 2016, 10 (2) : 413-427. doi: 10.3934/amc.2016015
[19]	John Sheekey. A new family of linear maximum rank distance codes. Advances in Mathematics of Communications, 2016, 10 (3) : 475-488. doi: 10.3934/amc.2016019
[20]	Sobhan Seyfaddini. Unboundedness of the Lagrangian Hofer distance in the Euclidean ball. Electronic Research Announcements, 2014, 21: 1-7. doi: 10.3934/era.2014.21.1

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Xing Huang Feng-Yu Wang
on Google Scholar
Xing Huang Feng-Yu Wang

2021 Impact Factor: 1.588

Tools

Article outline

2021 Impact Factor: 1.588

Tools

Metrics

Other articles
by authors

on AIMS
Xing Huang Feng-Yu Wang
on Google Scholar
Xing Huang Feng-Yu Wang

[Back to Top]