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June  2019, 39(6): 3017-3035. doi: 10.3934/dcds.2019125

Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs

Xing Huang 1, , Michael Röckner 2, and  Feng-Yu Wang 1,3,,

1. 

Center for Applied Mathematics, Tianjin University, Tianjin 300072, China
2. 

Department of Mathematics, Bielefeld University, D-33501 Bielefeld, Germany
3. 

Department of Mathematics, Swansea University, Singleton Park, SA2 8PP, United Kingdom

* Corresponding author: Feng-Yu Wang

Received  January 2018 Revised  October 2018 Published  February 2019

Fund Project: Supported by NNSFC (11801406, 11771326, 11831014, 11431014, 11726627).

By investigating path-distribution dependent stochastic differential equations, the following type of nonlinear Fokker–Planck equations for probability measures
$ (\mu_t)_{t \geq 0} $
 on the path space
$ {\scr {C}}: = C([-r_0, 0];\mathbb R^d), $
 is analyzed:
$ \partial_t \mu(t) = L_{t, \mu_t}^*\mu_t, \ \ t\ge 0, $
where
$ \mu(t) $
 is the image of
$ \mu_t $
 under the projection
$ {\scr {C}}\ni\xi\mapsto \xi(0)\in\mathbb R^d $
, and
$ \begin{align*} L_{t, \mu}(\xi)&: = \frac 1 2\sum\limits_{i, j = 1}^d a_{ij}(t, \xi, \mu)\frac{\partial^2} {\partial_{\xi(0)_i} \partial_{\xi(0)_j }} \\\; &\quad +\sum\limits_{i = 1}^d b_i(t, \xi, \mu)\frac{\partial}{\partial_{\xi(0)_i}}, \ \ t\ge 0, \xi\in {\scr {C}}, \mu\in \scr P^{\scr {C}}. \end{align*} $
Under reasonable conditions on the coefficients
$ a_{ij} $
 and
$ b_i $
, the existence, uniqueness, Lipschitz continuity in Wasserstein distance, total variational norm and entropy, as well as derivative estimates are derived for the martingale solutions.
Keywords: Nonlinear PDE for probability measures, path-distribution dependent SDEs, Wasserstein distance, Harnack inequality, coupling by change of measure.
Mathematics Subject Classification: Primary: 60J75, 47G20, 60G52.
Citation: Xing Huang, Michael Röckner, Feng-Yu Wang. Nonlinear Fokker–Planck equations for probability measures on path space and path-distribution dependent SDEs. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3017-3035. doi: 10.3934/dcds.2019125
References:
[1]

M. Arnaudon, A. Thalmaier and F.-Y. Wang, Harnack inequality and heat kernel estimate on manifolds with curvature unbounded below, Bull. Sci. Math., 130 (2006), 223-233.  doi: 10.1016/j.bulsci.2005.10.001.  Google Scholar

[2]

J. Bao, F.-Y. Wang and C. Yuan, Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory, Stochastic Processes and their Applications, , 2018, arXiv: 1710.01042. doi: 10.1016/j.spa.2018.12.010.  Google Scholar

[3]

J. Bao, F.-Y. Wang and C. Yuan, Ergodicity for Neutral Type SDEs with Infinite Length of Memory, preprint, arXiv: 1805.03431. Google Scholar

[4]

V. Bogachev, A. Krylov, M. Röckner and S. Shaposhnikov, Fokker-Planck-Kolmogorov Equations, Monograph, AMS, 2015. doi: 10.1090/surv/207.  Google Scholar

[5]

O. A. Butkovsky, On ergodic properties of nonlinear Markov chains and stochastic McKean-Vlasov equations, Theo. Probab. Appl., 58 (2014), 661-674.  doi: 10.1137/S0040585X97986825.  Google Scholar

[6] I. Csiszár and J. Körne, Information Theory: Coding Theorems for Discrete Memory-less Systems, Academic Press, New York, 1981.   Google Scholar
[7]

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.  Google Scholar

[8]

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.  Google Scholar

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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.  Google Scholar

[10]

A. Eberle, A. Guillin and R. Zimmer, Quantitative Harris type theorems for diffusions and McKean-Vlasov processes, Trans. Amer. Math. Soc., 2018, arXiv: 1606.06012. doi: 10.1090/tran/7576.  Google Scholar

[11]

N. Fournier and A. Guillin, From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules, Ann. Sci. l'ENS, 50 (2017), 157-199.  doi: 10.24033/asens.2318.  Google Scholar

[12]

H. Guérin, Existence and regularity of a weak function-solution for some Landau equations with a stochastic approach, Stoch. Proc. Appl., 101 (2002), 303-325.  doi: 10.1016/S0304-4149(02)00107-2.  Google Scholar

[13]

M. Hairer, J. C. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris theorem with applications to stochastic delay equations, Probab. Theory Related Fields, 149 (2011), 223-259.  doi: 10.1007/s00440-009-0250-6.  Google Scholar

[14]

I. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Amsterdam, 1981.  Google Scholar

[15]

A. N. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann., 104 (1931), 415-458.  doi: 10.1007/BF01457949.  Google Scholar

[16]

Yu. S. Mishura and A. Yu. Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, preprint, arXiv: 1603.02212. Google Scholar

[17]

S.-E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984.  Google Scholar

[18]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964.  Google Scholar

[19]

M. Röckner and F.-Y. Wang, Log-harnack inequality for stochastic differential equations in Hilbert spaces and its consequences, Infin. Dim. Anal. Quat. Probab. Relat. Top., 13 (2010), 27-37.  doi: 10.1142/S0219025710003936.  Google Scholar

[20]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, New York, 1979.  Google Scholar

[21]

A.-S. Sznitman, Topics in propagation of chaos, Lecture Notes in Mathematics, 1464 (1991), 165-251.  doi: 10.1007/BFb0085169.  Google Scholar

[22]

C. Villani, Optimal Transport, Old and New, Springer-Verlg, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[23]

C. Villani, On the spatially homogeneous Landau equation for Maxwellian mocecules, Math. Mod. Meth. Appl. Sci., 8 (1998), 957-983.  doi: 10.1142/S0218202598000433.  Google Scholar

[24]

F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields, 109 (1997), 417-424.  doi: 10.1007/s004400050137.  Google Scholar

[25]

F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab., 35 (2007), 1333-1350.  doi: 10.1214/009117906000001204.  Google Scholar

[26]

F.-Y. Wang, Harnack Inequalities and Applications for Stochastic Partial Differential Equations, Springer, Berlin, 2013. doi: 10.1007/978-1-4614-7934-5.  Google Scholar

[27]

F.-Y. Wang, Harnack inequalities on manifolds with boundary and applications, J. Math. Pures Appl., 94 (2010), 304-321.  doi: 10.1016/j.matpur.2010.03.001.  Google Scholar

[28]

F.-Y. Wang, Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds, Ann. Probab., 39 (2011), 1449-1467.  doi: 10.1214/10-AOP600.  Google Scholar

[29]

F.-Y. Wang, Integration by parts formula and shift Harnack inequality for stochastic equations, Ann. Probab., 42 (2014), 994-1019.  doi: 10.1214/13-AOP875.  Google Scholar

[30]

F.-Y. Wang and C. Yuan, Harnack inequalities for functional SDEs with multiplicative noise and applications, Stoch. Proc. Appl., 121 (2011), 2692-2710.  doi: 10.1016/j.spa.2011.07.001.  Google Scholar

[31]

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.  Google Scholar

[32]

T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations Ⅰ & Ⅱ, J. Math. Kyoto Univ., 11 (1971), 155-167.  doi: 10.1215/kjm/1250523620.  Google Scholar

show all references

References:
[1]

M. Arnaudon, A. Thalmaier and F.-Y. Wang, Harnack inequality and heat kernel estimate on manifolds with curvature unbounded below, Bull. Sci. Math., 130 (2006), 223-233.  doi: 10.1016/j.bulsci.2005.10.001.  Google Scholar

[2]

J. Bao, F.-Y. Wang and C. Yuan, Asymptotic Log-Harnack inequality and applications for stochastic systems of infinite memory, Stochastic Processes and their Applications, , 2018, arXiv: 1710.01042. doi: 10.1016/j.spa.2018.12.010.  Google Scholar

[3]

J. Bao, F.-Y. Wang and C. Yuan, Ergodicity for Neutral Type SDEs with Infinite Length of Memory, preprint, arXiv: 1805.03431. Google Scholar

[4]

V. Bogachev, A. Krylov, M. Röckner and S. Shaposhnikov, Fokker-Planck-Kolmogorov Equations, Monograph, AMS, 2015. doi: 10.1090/surv/207.  Google Scholar

[5]

O. A. Butkovsky, On ergodic properties of nonlinear Markov chains and stochastic McKean-Vlasov equations, Theo. Probab. Appl., 58 (2014), 661-674.  doi: 10.1137/S0040585X97986825.  Google Scholar

[6] I. Csiszár and J. Körne, Information Theory: Coding Theorems for Discrete Memory-less Systems, Academic Press, New York, 1981.   Google Scholar
[7]

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.  Google Scholar

[8]

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.  Google Scholar

[9]

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.  Google Scholar

[10]

A. Eberle, A. Guillin and R. Zimmer, Quantitative Harris type theorems for diffusions and McKean-Vlasov processes, Trans. Amer. Math. Soc., 2018, arXiv: 1606.06012. doi: 10.1090/tran/7576.  Google Scholar

[11]

N. Fournier and A. Guillin, From a Kac-like particle system to the Landau equation for hard potentials and Maxwell molecules, Ann. Sci. l'ENS, 50 (2017), 157-199.  doi: 10.24033/asens.2318.  Google Scholar

[12]

H. Guérin, Existence and regularity of a weak function-solution for some Landau equations with a stochastic approach, Stoch. Proc. Appl., 101 (2002), 303-325.  doi: 10.1016/S0304-4149(02)00107-2.  Google Scholar

[13]

M. Hairer, J. C. Mattingly and M. Scheutzow, Asymptotic coupling and a general form of Harris theorem with applications to stochastic delay equations, Probab. Theory Related Fields, 149 (2011), 223-259.  doi: 10.1007/s00440-009-0250-6.  Google Scholar

[14]

I. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Amsterdam, 1981.  Google Scholar

[15]

A. N. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann., 104 (1931), 415-458.  doi: 10.1007/BF01457949.  Google Scholar

[16]

Yu. S. Mishura and A. Yu. Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, preprint, arXiv: 1603.02212. Google Scholar

[17]

S.-E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984.  Google Scholar

[18]

M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964.  Google Scholar

[19]

M. Röckner and F.-Y. Wang, Log-harnack inequality for stochastic differential equations in Hilbert spaces and its consequences, Infin. Dim. Anal. Quat. Probab. Relat. Top., 13 (2010), 27-37.  doi: 10.1142/S0219025710003936.  Google Scholar

[20]

D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, New York, 1979.  Google Scholar

[21]

A.-S. Sznitman, Topics in propagation of chaos, Lecture Notes in Mathematics, 1464 (1991), 165-251.  doi: 10.1007/BFb0085169.  Google Scholar

[22]

C. Villani, Optimal Transport, Old and New, Springer-Verlg, Berlin, 2009. doi: 10.1007/978-3-540-71050-9.  Google Scholar

[23]

C. Villani, On the spatially homogeneous Landau equation for Maxwellian mocecules, Math. Mod. Meth. Appl. Sci., 8 (1998), 957-983.  doi: 10.1142/S0218202598000433.  Google Scholar

[24]

F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields, 109 (1997), 417-424.  doi: 10.1007/s004400050137.  Google Scholar

[25]

F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab., 35 (2007), 1333-1350.  doi: 10.1214/009117906000001204.  Google Scholar

[26]

F.-Y. Wang, Harnack Inequalities and Applications for Stochastic Partial Differential Equations, Springer, Berlin, 2013. doi: 10.1007/978-1-4614-7934-5.  Google Scholar

[27]

F.-Y. Wang, Harnack inequalities on manifolds with boundary and applications, J. Math. Pures Appl., 94 (2010), 304-321.  doi: 10.1016/j.matpur.2010.03.001.  Google Scholar

[28]

F.-Y. Wang, Harnack inequality for SDE with multiplicative noise and extension to Neumann semigroup on nonconvex manifolds, Ann. Probab., 39 (2011), 1449-1467.  doi: 10.1214/10-AOP600.  Google Scholar

[29]

F.-Y. Wang, Integration by parts formula and shift Harnack inequality for stochastic equations, Ann. Probab., 42 (2014), 994-1019.  doi: 10.1214/13-AOP875.  Google Scholar

[30]

F.-Y. Wang and C. Yuan, Harnack inequalities for functional SDEs with multiplicative noise and applications, Stoch. Proc. Appl., 121 (2011), 2692-2710.  doi: 10.1016/j.spa.2011.07.001.  Google Scholar

[31]

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.  Google Scholar

[32]

T. Yamada and S. Watanabe, On the uniqueness of solutions of stochastic differential equations Ⅰ & Ⅱ, J. Math. Kyoto Univ., 11 (1971), 155-167.  doi: 10.1215/kjm/1250523620.  Google Scholar

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