American Institute of Mathematical Sciences

June  2021, 20(6): 2399-2419. doi: 10.3934/cpaa.2021087

Density functions of distribution dependent SDEs driven by Lévy noises

 Department of Mathematics, Nanjing University, Nanjing, 210093, China

Received  November 2020 Revised  April 2021 Published  June 2021 Early access  May 2021

Fund Project: This work is supported by NNSFC (No. 11971227, 11790272)

By Malliavin calculus for Wiener-Poisson functionals and Lions derivative for probability measures, existence and smoothness of density functions for distribution dependent SDEs with Lévy noises are derived.

Citation: Yulin Song. Density functions of distribution dependent SDEs driven by Lévy noises. Communications on Pure & Applied Analysis, 2021, 20 (6) : 2399-2419. doi: 10.3934/cpaa.2021087
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] J. M. Bismut, Calcul des variations stochastiques et processus de sauts, Z. Wahrsch. Verw. Gebiete, 63 (1983), 147–235. doi: 10.1007/BF00538963.  Google Scholar [4] W. Brown and K. Hepp, The Vlasov dynamics and its fluctuation in the $1/N$ limit of interacting particles, Commun. Math. Phys., 66 (1977), 101–113.  Google Scholar [5] R. Buckdahn, J. Li, S. Peng and C. Rainer, Mean-field stochastic differential equations and associated PDEs, Ann. Probab., 45 (2017), 824–878. doi: 10.1214/15-AOP1076.  Google Scholar [6] P. Cattiaux and L. Mesnager, Hypoelliptic non-homogeneous diffusions, Probab. Theory Related Fields, 123 (2002), 453–483. doi: 10.1007/s004400100194.  Google Scholar [7] P. Cardaliaguet, Notes on Mean Filed Games, from P. L. Lions' lectures at Collège de France, 2013. Google Scholar [8] 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.  Google Scholar [9] 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.  Google Scholar [10] R. Höpfner, E. Löcherbach and M. Thieullen, Strongly degenerate time inhomogeneous SDEs: Densities and support properties. Application to Hodgkin-Huxley type systems, Bernoulli, 23 (2017), 2587–2616. doi: 10.3150/16-BEJ820.  Google Scholar [11] 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 [12] X. Huang and Y. Song, Well-posedness and regularity for distribution dependent SPDEs with singular drifts, Nonlinear Anal., 203 (2021), 112167. doi: 10.1016/j.na.2020.112167.  Google Scholar [13] 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.  Google Scholar [14] H. Kunita, Nondegenerate SDE's with jumps and their hypoelliptic properties, J. Math. Soc. Japan, 65 (2013), 993–1035.  Google Scholar [15] P. Lions, Cours au Collège de France: Théorie des jeuàchamps moyens, available from: http://www.college-de-france.fr/default/EN/all/equ[1]der/audiovideo.jsp. Google Scholar [16] H. P. McKean, Propagation of chaos for a class of nonlinear parabolic equations, Lecture Series in Differential Equations, 7 (1967), 41–57.  Google Scholar [17] D. Nualart, The Malliavin Calsulus and Related Topics, 2$^nd$ edition, Springer-Verlag, New York, 2006.  Google Scholar [18] E. Priola and J. Zabczyk, Densities for Ornstein-Uhlenbeck processes with jumps, Bull. Lond. Math. Soc., 41 (2009), 41-50.  doi: 10.1112/blms/bdn099.  Google Scholar [19] P. Ren and F. Y. Wang, Bismut formula for Lions derivative of distribution dependent SDEs and applications, J. Differ. Equ., 267 (2019), 4745-4777.  doi: 10.1016/j.jde.2019.05.016.  Google Scholar [20] M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, Bernoulli, 27 (2021), 1131-1158.   Google Scholar [21] 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.  Google Scholar [22] Y. Song and Y. Xie, Existence of density functions for the running maximum of a Lévy-Itô diffusion, Potential Anal., 48 (2018), 35-48.  doi: 10.1007/s11118-017-9625-y.  Google Scholar [23] Y. Song and X. Zhang, Regularity of density for SDEs driven by degenrate Lévy noises, Electron. J. Probab., 20 (2015), 1-27.  doi: 10.1214/EJP.v20-3287.  Google Scholar [24] S. Taniguchi, Applications of Malliavin's calculus to time-dependent systems of heat equations, Osaka J. Math., 22 (1985), 307-320.   Google Scholar [25] 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.  Google Scholar [26] X. Zhang, Densities for SDEs driven by degenerate $\alpha$-stable processes, Ann. Probab., 42 (2014), 1885-1910.  doi: 10.1214/13-AOP900.  Google Scholar

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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] J. M. Bismut, Calcul des variations stochastiques et processus de sauts, Z. Wahrsch. Verw. Gebiete, 63 (1983), 147–235. doi: 10.1007/BF00538963.  Google Scholar [4] W. Brown and K. Hepp, The Vlasov dynamics and its fluctuation in the $1/N$ limit of interacting particles, Commun. Math. Phys., 66 (1977), 101–113.  Google Scholar [5] R. Buckdahn, J. Li, S. Peng and C. Rainer, Mean-field stochastic differential equations and associated PDEs, Ann. Probab., 45 (2017), 824–878. doi: 10.1214/15-AOP1076.  Google Scholar [6] P. Cattiaux and L. Mesnager, Hypoelliptic non-homogeneous diffusions, Probab. Theory Related Fields, 123 (2002), 453–483. doi: 10.1007/s004400100194.  Google Scholar [7] P. Cardaliaguet, Notes on Mean Filed Games, from P. L. Lions' lectures at Collège de France, 2013. Google Scholar [8] 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.  Google Scholar [9] 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.  Google Scholar [10] R. Höpfner, E. Löcherbach and M. Thieullen, Strongly degenerate time inhomogeneous SDEs: Densities and support properties. Application to Hodgkin-Huxley type systems, Bernoulli, 23 (2017), 2587–2616. doi: 10.3150/16-BEJ820.  Google Scholar [11] 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 [12] X. Huang and Y. Song, Well-posedness and regularity for distribution dependent SPDEs with singular drifts, Nonlinear Anal., 203 (2021), 112167. doi: 10.1016/j.na.2020.112167.  Google Scholar [13] 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.  Google Scholar [14] H. Kunita, Nondegenerate SDE's with jumps and their hypoelliptic properties, J. Math. Soc. Japan, 65 (2013), 993–1035.  Google Scholar [15] P. Lions, Cours au Collège de France: Théorie des jeuàchamps moyens, available from: http://www.college-de-france.fr/default/EN/all/equ[1]der/audiovideo.jsp. Google Scholar [16] H. P. McKean, Propagation of chaos for a class of nonlinear parabolic equations, Lecture Series in Differential Equations, 7 (1967), 41–57.  Google Scholar [17] D. Nualart, The Malliavin Calsulus and Related Topics, 2$^nd$ edition, Springer-Verlag, New York, 2006.  Google Scholar [18] E. Priola and J. Zabczyk, Densities for Ornstein-Uhlenbeck processes with jumps, Bull. Lond. Math. Soc., 41 (2009), 41-50.  doi: 10.1112/blms/bdn099.  Google Scholar [19] P. Ren and F. Y. Wang, Bismut formula for Lions derivative of distribution dependent SDEs and applications, J. Differ. Equ., 267 (2019), 4745-4777.  doi: 10.1016/j.jde.2019.05.016.  Google Scholar [20] M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, Bernoulli, 27 (2021), 1131-1158.   Google Scholar [21] 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.  Google Scholar [22] Y. Song and Y. Xie, Existence of density functions for the running maximum of a Lévy-Itô diffusion, Potential Anal., 48 (2018), 35-48.  doi: 10.1007/s11118-017-9625-y.  Google Scholar [23] Y. Song and X. Zhang, Regularity of density for SDEs driven by degenrate Lévy noises, Electron. J. Probab., 20 (2015), 1-27.  doi: 10.1214/EJP.v20-3287.  Google Scholar [24] S. Taniguchi, Applications of Malliavin's calculus to time-dependent systems of heat equations, Osaka J. Math., 22 (1985), 307-320.   Google Scholar [25] 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.  Google Scholar [26] X. Zhang, Densities for SDEs driven by degenerate $\alpha$-stable processes, Ann. Probab., 42 (2014), 1885-1910.  doi: 10.1214/13-AOP900.  Google Scholar
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