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 and 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.

[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]

J. M. Bismut, Calcul des variations stochastiques et processus de sauts, Z. Wahrsch. Verw. Gebiete, 63 (1983), 147–235. doi: 10.1007/BF00538963.

[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.

[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.

[6]

P. Cattiaux and L. Mesnager, Hypoelliptic non-homogeneous diffusions, Probab. Theory Related Fields, 123 (2002), 453–483. doi: 10.1007/s004400100194.

[7]

P. Cardaliaguet, Notes on Mean Filed Games, from P. L. Lions' lectures at Collège de France, 2013.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

H. Kunita, Nondegenerate SDE's with jumps and their hypoelliptic properties, J. Math. Soc. Japan, 65 (2013), 993–1035.

[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.

[16]

H. P. McKean, Propagation of chaos for a class of nonlinear parabolic equations, Lecture Series in Differential Equations, 7 (1967), 41–57.

[17]

D. Nualart, The Malliavin Calsulus and Related Topics, 2$^nd$ edition, Springer-Verlag, New York, 2006.

[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.

[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.

[20]

M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, Bernoulli, 27 (2021), 1131-1158. 

[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.

[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.

[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.

[24]

S. Taniguchi, Applications of Malliavin's calculus to time-dependent systems of heat equations, Osaka J. Math., 22 (1985), 307-320. 

[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.

[26]

X. Zhang, Densities for SDEs driven by degenerate $\alpha$-stable processes, Ann. Probab., 42 (2014), 1885-1910.  doi: 10.1214/13-AOP900.

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]

J. M. Bismut, Calcul des variations stochastiques et processus de sauts, Z. Wahrsch. Verw. Gebiete, 63 (1983), 147–235. doi: 10.1007/BF00538963.

[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.

[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.

[6]

P. Cattiaux and L. Mesnager, Hypoelliptic non-homogeneous diffusions, Probab. Theory Related Fields, 123 (2002), 453–483. doi: 10.1007/s004400100194.

[7]

P. Cardaliaguet, Notes on Mean Filed Games, from P. L. Lions' lectures at Collège de France, 2013.

[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.

[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.

[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.

[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.

[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.

[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.

[14]

H. Kunita, Nondegenerate SDE's with jumps and their hypoelliptic properties, J. Math. Soc. Japan, 65 (2013), 993–1035.

[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.

[16]

H. P. McKean, Propagation of chaos for a class of nonlinear parabolic equations, Lecture Series in Differential Equations, 7 (1967), 41–57.

[17]

D. Nualart, The Malliavin Calsulus and Related Topics, 2$^nd$ edition, Springer-Verlag, New York, 2006.

[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.

[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.

[20]

M. Röckner and X. Zhang, Well-posedness of distribution dependent SDEs with singular drifts, Bernoulli, 27 (2021), 1131-1158. 

[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.

[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.

[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.

[24]

S. Taniguchi, Applications of Malliavin's calculus to time-dependent systems of heat equations, Osaka J. Math., 22 (1985), 307-320. 

[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.

[26]

X. Zhang, Densities for SDEs driven by degenerate $\alpha$-stable processes, Ann. Probab., 42 (2014), 1885-1910.  doi: 10.1214/13-AOP900.

[1]

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  doi: 10.3934/dcds.2022065

[2]

Adam Andersson, Felix Lindner. Malliavin regularity and weak approximation of semilinear SPDEs with Lévy noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4271-4294. doi: 10.3934/dcdsb.2019081

[3]

Chaman Kumar, Sotirios Sabanis. On tamed milstein schemes of SDEs driven by Lévy noise. Discrete and Continuous Dynamical Systems - B, 2017, 22 (2) : 421-463. doi: 10.3934/dcdsb.2017020

[4]

Ziheng Chen, Siqing Gan, Xiaojie Wang. Mean-square approximations of Lévy noise driven SDEs with super-linearly growing diffusion and jump coefficients. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4513-4545. doi: 10.3934/dcdsb.2019154

[5]

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

[6]

Badr-eddine Berrhazi, Mohamed El Fatini, Tomás Caraballo, Roger Pettersson. A stochastic SIRI epidemic model with Lévy noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (6) : 2415-2431. doi: 10.3934/dcdsb.2018057

[7]

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

[8]

Linghua Chen, Espen R. Jakobsen. L1 semigroup generation for Fokker-Planck operators associated to general Lévy driven SDEs. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5735-5763. doi: 10.3934/dcds.2018250

[9]

Caroline Hillairet, Ying Jiao, Anthony Réveillac. Pricing formulae for derivatives in insurance using Malliavin calculus. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 7-. doi: 10.1186/s41546-018-0028-9

[10]

Xiangjun Wang, Jianghui Wen, Jianping Li, Jinqiao Duan. Impact of $\alpha$-stable Lévy noise on the Stommel model for the thermohaline circulation. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1575-1584. doi: 10.3934/dcdsb.2012.17.1575

[11]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[12]

Kexue Li, Jigen Peng, Junxiong Jia. Explosive solutions of parabolic stochastic partial differential equations with lévy noise. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5105-5125. doi: 10.3934/dcds.2017221

[13]

Justin Cyr, Phuong Nguyen, Sisi Tang, Roger Temam. Review of local and global existence results for stochastic pdes with Lévy noise. Discrete and Continuous Dynamical Systems, 2020, 40 (10) : 5639-5710. doi: 10.3934/dcds.2020241

[14]

Justin Cyr, Phuong Nguyen, Roger Temam. Stochastic one layer shallow water equations with Lévy noise. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 3765-3818. doi: 10.3934/dcdsb.2018331

[15]

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

[16]

Chaman Kumar. On Milstein-type scheme for SDE driven by Lévy noise with super-linear coefficients. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1405-1446. doi: 10.3934/dcdsb.2020167

[17]

Xueqin Li, Chao Tang, Tianmin Huang. Poisson $S^2$-almost automorphy for stochastic processes and its applications to SPDEs driven by Lévy noise. Discrete and Continuous Dynamical Systems - B, 2018, 23 (8) : 3309-3345. doi: 10.3934/dcdsb.2018282

[18]

Markus Riedle, Jianliang Zhai. Large deviations for stochastic heat equations with memory driven by Lévy-type noise. Discrete and Continuous Dynamical Systems, 2018, 38 (4) : 1983-2005. doi: 10.3934/dcds.2018080

[19]

Phuong Nguyen, Roger Temam. The stampacchia maximum principle for stochastic partial differential equations forced by lévy noise. Communications on Pure and Applied Analysis, 2020, 19 (4) : 2289-2331. doi: 10.3934/cpaa.2020100

[20]

Kumarasamy Sakthivel, Sivaguru S. Sritharan. Martingale solutions for stochastic Navier-Stokes equations driven by Lévy noise. Evolution Equations and Control Theory, 2012, 1 (2) : 355-392. doi: 10.3934/eect.2012.1.355

2020 Impact Factor: 1.916

Metrics

  • PDF downloads (160)
  • HTML views (136)
  • Cited by (0)

Other articles
by authors

[Back to Top]