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Bistable reaction equations with doubly nonlinear diffusion
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 |
$ (\mu_t)_{t \geq 0} $ |
$ {\scr {C}}: = C([-r_0, 0];\mathbb R^d), $ |
$ \partial_t \mu(t) = L_{t, \mu_t}^*\mu_t, \ \ t\ge 0, $ |
$ \mu(t) $ |
$ \mu_t $ |
$ {\scr {C}}\ni\xi\mapsto \xi(0)\in\mathbb R^d $ |
$ \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*} $ |
$ a_{ij} $ |
$ b_i $ |
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. |
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. |
[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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