# American Institute of Mathematical Sciences

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

 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.
Citation: 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
##### 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. [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. [3] J. Bao, F.-Y. Wang and C. Yuan, Ergodicity for Neutral Type SDEs with Infinite Length of Memory, preprint, arXiv: 1805.03431. [4] V. Bogachev, A. Krylov, M. Röckner and S. Shaposhnikov, Fokker-Planck-Kolmogorov Equations, Monograph, AMS, 2015. doi: 10.1090/surv/207. [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. [6] I. Csiszár and J. Körne, Information Theory: Coding Theorems for Discrete Memory-less Systems, Academic Press, New York, 1981. [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. [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. [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. [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. [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. [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. [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. [14] I. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Amsterdam, 1981. [15] A. N. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann., 104 (1931), 415-458.  doi: 10.1007/BF01457949. [16] Yu. S. Mishura and A. Yu. Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, preprint, arXiv: 1603.02212. [17] S.-E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984. [18] M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964. [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. [20] D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, New York, 1979. [21] A.-S. Sznitman, Topics in propagation of chaos, Lecture Notes in Mathematics, 1464 (1991), 165-251.  doi: 10.1007/BFb0085169. [22] C. Villani, Optimal Transport, Old and New, Springer-Verlg, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. [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. [24] F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields, 109 (1997), 417-424.  doi: 10.1007/s004400050137. [25] F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab., 35 (2007), 1333-1350.  doi: 10.1214/009117906000001204. [26] F.-Y. Wang, Harnack Inequalities and Applications for Stochastic Partial Differential Equations, Springer, Berlin, 2013. doi: 10.1007/978-1-4614-7934-5. [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. [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. [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. [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. [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. [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.

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##### 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. [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. [3] J. Bao, F.-Y. Wang and C. Yuan, Ergodicity for Neutral Type SDEs with Infinite Length of Memory, preprint, arXiv: 1805.03431. [4] V. Bogachev, A. Krylov, M. Röckner and S. Shaposhnikov, Fokker-Planck-Kolmogorov Equations, Monograph, AMS, 2015. doi: 10.1090/surv/207. [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. [6] I. Csiszár and J. Körne, Information Theory: Coding Theorems for Discrete Memory-less Systems, Academic Press, New York, 1981. [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. [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. [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. [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. [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. [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. [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. [14] I. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, North-Holland Publishing Co., Amsterdam, 1981. [15] A. N. Kolmogoroff, Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung, Math. Ann., 104 (1931), 415-458.  doi: 10.1007/BF01457949. [16] Yu. S. Mishura and A. Yu. Veretennikov, Existence and uniqueness theorems for solutions of McKean-Vlasov stochastic equations, preprint, arXiv: 1603.02212. [17] S.-E. A. Mohammed, Stochastic Functional Differential Equations, Pitman, Boston, 1984. [18] M. S. Pinsker, Information and Information Stability of Random Variables and Processes, Holden-Day, San Francisco, 1964. [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. [20] D. W. Stroock and S. R. S. Varadhan, Multidimensional Diffusion Processes, Springer, New York, 1979. [21] A.-S. Sznitman, Topics in propagation of chaos, Lecture Notes in Mathematics, 1464 (1991), 165-251.  doi: 10.1007/BFb0085169. [22] C. Villani, Optimal Transport, Old and New, Springer-Verlg, Berlin, 2009. doi: 10.1007/978-3-540-71050-9. [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. [24] F.-Y. Wang, Logarithmic Sobolev inequalities on noncompact Riemannian manifolds, Probab. Theory Related Fields, 109 (1997), 417-424.  doi: 10.1007/s004400050137. [25] F.-Y. Wang, Harnack inequality and applications for stochastic generalized porous media equations, Ann. Probab., 35 (2007), 1333-1350.  doi: 10.1214/009117906000001204. [26] F.-Y. Wang, Harnack Inequalities and Applications for Stochastic Partial Differential Equations, Springer, Berlin, 2013. doi: 10.1007/978-1-4614-7934-5. [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. [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. [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. [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. [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. [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.
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