doi: 10.3934/dcds.2022065
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Bismut formula for intrinsic/Lions derivatives of distribution dependent SDEs with singular coefficients

1. 

Center for Applied Mathematics, Tianjin University, Tianjin, China

2. 

Department of Mathematics, Nanjing University, Nanjing, China

3. 

Center for Applied Mathematics, Tianjin University, Tianjin, China, Department of Mathematics, Swansea University, Bay Campus, Swansea, SA1 8EN, United Kingdom

*Corresponding author: Yulin Song

Received  September 2021 Revised  March 2022 Early access May 2022

Fund Project: The first author is supported by NNSFC (No.11801406), the second author is supported by NNSFC (No.11971227, 11790272) and the third author is supported by NNSFC (No.11771326, 11831014, 11921001)

By using distribution dependent Zvonkin's transforms and Malliavin calculus, the Bismut type formula is derived for the intrinisc/Lions derivatives of distribution dependent SDEs with singular drifts, which generalizes the corresponding results derived for classical SDEs and regular distribution dependent SDEs.

Citation: 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, doi: 10.3934/dcds.2022065
References:
[1]

S. AlbeverioY. G. Kondratiev and M. Röckner, Differential geometry of Poisson spaces, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 1129-1134. 

[2]

D. Baños, The Bismut-Elworthy-Li formula for mean-field stochastic differential equations, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 220-233.  doi: 10.1214/16-AIHP801.

[3]

J. BaoP. Ren and F.-Y. Wang, Bismut formulas for Lions derivative of McKean-Vlasov SDEs with memory, J. Differential Equations, 282 (2021), 285-329.  doi: 10.1016/j.jde.2021.02.019.

[4]

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.

[5]

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.

[6]

S. BenachourB. RoynetteD. Talay and P. Vallois, Nonlinear selfstabilizing processes. I. Existence, invariant probability, propagation of chaos, Stochastic Process. Appl., 75 (1998), 173-201.  doi: 10.1016/S0304-4149(98)00018-0.

[7]

J. M. Bismut, Large Deviations and the Malliavin Calculus, Birkhäser, Boston, 1984.

[8]

M. Bossy and D. Talay, A stochastic particle method for the McKean-Vlasov and the Burgers equation, Math. Comp., 66 (1997), 157-192.  doi: 10.1090/S0025-5718-97-00776-X.

[9]

P. Cardaliaguet, Notes on Mean Filed Games, 2013. Available from: https://www.ceremade.dauphine.fr/cardaliaguet/MFG20130420.pdf.

[10]

R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.  doi: 10.1137/120883499.

[11]

R. Carmona and F. Delarue, Forward-backward stochastic differential equations and controlled McKean-Vlasov dynamics,, Ann. Probab., 43 (2015), 2647-2700.  doi: 10.1214/14-AOP946.

[12]

P.-E. Chaudry De Raynal and N. Frikha, Well-posedness for some non-linear SDEs and related PDE on the Wasserstein space, J. Math. Pures Appl., 159 (2022), 1-167.  doi: 10.1016/j.matpur.2021.12.001.

[13]

D. Crisan and E. McMurray, Smoothing properties of McKean-Vlasov SDEs, Probab. Theory Relat. Fields., 171 (2018), 97-148.  doi: 10.1007/s00440-017-0774-0.

[14]

D. Dawson and J. Vaillancourt, Stochastic McKean-Vlasov equations, Nonlinear Differential Equations Appl., 2 (1995), 199-229.  doi: 10.1007/BF01295311.

[15]

K. D. Elworthy and X.-M. Li, Formulae for the derivatives of the heat semigroups, J. Funct. Anal., 125 (1994), 252-286.  doi: 10.1006/jfan.1994.1124.

[16]

X. HuangP. Ren and F.-Y. Wang, Distribution dependent stochastic differential equations, Front. Math. China, 16 (2021), 257-301.  doi: 10.1007/s11464-021-0920-y.

[17]

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.

[18]

X. Huang and F.-Y. Wang, Distribution dependent SDEs with singular coefficients, Stochastic Process Appl., 129 (2019), 4747-4770.  doi: 10.1016/j.spa.2018.12.012.

[19]

X. Huang and F.-Y. Wang, McKean-Vlasov SDEs with drifts discontinuous under Wasserstein distance, Discrete Contin. Dyn. Syst., 41 (2021), 1667-1679.  doi: 10.3934/dcds.2020336.

[20]

P. Kotelenez and T. Kurtz, Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type, Probab. Theory Related Fields, 146 (2010), 189-222.  doi: 10.1007/s00440-008-0188-0.

[21]

Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, Theor. Probability and Math. Statist., 103 (2020), 59-101. 

[22]

D. Nualart, The Malliavin Calculus and Related Topics, 2$^{nd}$ edition, Springer-Verlag, Berlin Heidelberg, 2006.

[23]

P. Ren and F.-Y. Wang, Bismut formula for Lions derivative of distribution dependent SDEs and applications, J. Differential Equations, 267 (2019), 4745-4777.  doi: 10.1016/j.jde.2019.05.016.

[24]

A.-S. Sznitman, Topics in propagation of chaos, In École d'Été de Probabilités de Sain-Flour XIX-1989, Lecture Notes in Mathematics, Springer, Berlin, 1464 (1991), 165–251. doi: 10.1007/BFb0085169.

[25]

Y. Song, Gradient estimates and exponential ergodicity for mean-field SDEs with jumps, J. Theoret. Probab., 33 (2020), 201-238.  doi: 10.1007/s10959-018-0845-x.

[26]

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.

[27]

F.-Y. Wang, Distribution-dependent SDEs for Landau type equations, Stochastic. Process. Appl., 128 (2018), 595-621.  doi: 10.1016/j.spa.2017.05.006.

[28]

P. XiaL. XieX. Zhang and G. Zhao, $L^q$($L^p$)-theory of stochastic differential equations, Stochastic Process. Appl., 130 (2020), 5188-5211.  doi: 10.1016/j.spa.2020.03.004.

[29]

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]

S. AlbeverioY. G. Kondratiev and M. Röckner, Differential geometry of Poisson spaces, C. R. Acad. Sci. Paris Sér. I Math., 323 (1996), 1129-1134. 

[2]

D. Baños, The Bismut-Elworthy-Li formula for mean-field stochastic differential equations, Ann. Inst. Henri Poincaré Probab. Stat., 54 (2018), 220-233.  doi: 10.1214/16-AIHP801.

[3]

J. BaoP. Ren and F.-Y. Wang, Bismut formulas for Lions derivative of McKean-Vlasov SDEs with memory, J. Differential Equations, 282 (2021), 285-329.  doi: 10.1016/j.jde.2021.02.019.

[4]

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.

[5]

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.

[6]

S. BenachourB. RoynetteD. Talay and P. Vallois, Nonlinear selfstabilizing processes. I. Existence, invariant probability, propagation of chaos, Stochastic Process. Appl., 75 (1998), 173-201.  doi: 10.1016/S0304-4149(98)00018-0.

[7]

J. M. Bismut, Large Deviations and the Malliavin Calculus, Birkhäser, Boston, 1984.

[8]

M. Bossy and D. Talay, A stochastic particle method for the McKean-Vlasov and the Burgers equation, Math. Comp., 66 (1997), 157-192.  doi: 10.1090/S0025-5718-97-00776-X.

[9]

P. Cardaliaguet, Notes on Mean Filed Games, 2013. Available from: https://www.ceremade.dauphine.fr/cardaliaguet/MFG20130420.pdf.

[10]

R. Carmona and F. Delarue, Probabilistic analysis of mean-field games, SIAM J. Control Optim., 51 (2013), 2705-2734.  doi: 10.1137/120883499.

[11]

R. Carmona and F. Delarue, Forward-backward stochastic differential equations and controlled McKean-Vlasov dynamics,, Ann. Probab., 43 (2015), 2647-2700.  doi: 10.1214/14-AOP946.

[12]

P.-E. Chaudry De Raynal and N. Frikha, Well-posedness for some non-linear SDEs and related PDE on the Wasserstein space, J. Math. Pures Appl., 159 (2022), 1-167.  doi: 10.1016/j.matpur.2021.12.001.

[13]

D. Crisan and E. McMurray, Smoothing properties of McKean-Vlasov SDEs, Probab. Theory Relat. Fields., 171 (2018), 97-148.  doi: 10.1007/s00440-017-0774-0.

[14]

D. Dawson and J. Vaillancourt, Stochastic McKean-Vlasov equations, Nonlinear Differential Equations Appl., 2 (1995), 199-229.  doi: 10.1007/BF01295311.

[15]

K. D. Elworthy and X.-M. Li, Formulae for the derivatives of the heat semigroups, J. Funct. Anal., 125 (1994), 252-286.  doi: 10.1006/jfan.1994.1124.

[16]

X. HuangP. Ren and F.-Y. Wang, Distribution dependent stochastic differential equations, Front. Math. China, 16 (2021), 257-301.  doi: 10.1007/s11464-021-0920-y.

[17]

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.

[18]

X. Huang and F.-Y. Wang, Distribution dependent SDEs with singular coefficients, Stochastic Process Appl., 129 (2019), 4747-4770.  doi: 10.1016/j.spa.2018.12.012.

[19]

X. Huang and F.-Y. Wang, McKean-Vlasov SDEs with drifts discontinuous under Wasserstein distance, Discrete Contin. Dyn. Syst., 41 (2021), 1667-1679.  doi: 10.3934/dcds.2020336.

[20]

P. Kotelenez and T. Kurtz, Macroscopic limits for stochastic partial differential equations of McKean-Vlasov type, Probab. Theory Related Fields, 146 (2010), 189-222.  doi: 10.1007/s00440-008-0188-0.

[21]

Y. S. Mishura and A. Y. Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, Theor. Probability and Math. Statist., 103 (2020), 59-101. 

[22]

D. Nualart, The Malliavin Calculus and Related Topics, 2$^{nd}$ edition, Springer-Verlag, Berlin Heidelberg, 2006.

[23]

P. Ren and F.-Y. Wang, Bismut formula for Lions derivative of distribution dependent SDEs and applications, J. Differential Equations, 267 (2019), 4745-4777.  doi: 10.1016/j.jde.2019.05.016.

[24]

A.-S. Sznitman, Topics in propagation of chaos, In École d'Été de Probabilités de Sain-Flour XIX-1989, Lecture Notes in Mathematics, Springer, Berlin, 1464 (1991), 165–251. doi: 10.1007/BFb0085169.

[25]

Y. Song, Gradient estimates and exponential ergodicity for mean-field SDEs with jumps, J. Theoret. Probab., 33 (2020), 201-238.  doi: 10.1007/s10959-018-0845-x.

[26]

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.

[27]

F.-Y. Wang, Distribution-dependent SDEs for Landau type equations, Stochastic. Process. Appl., 128 (2018), 595-621.  doi: 10.1016/j.spa.2017.05.006.

[28]

P. XiaL. XieX. Zhang and G. Zhao, $L^q$($L^p$)-theory of stochastic differential equations, Stochastic Process. Appl., 130 (2020), 5188-5211.  doi: 10.1016/j.spa.2020.03.004.

[29]

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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