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  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

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.  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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