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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 |
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 [
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. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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. 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. |
[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. Google Scholar |
[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. 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. |
[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. 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. |
[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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