Dynamical systems in the variational formulation of the Fokker-Planck equation by the Wasserstein metric

Applied Mathematics and Optimization

Volumn 42, Issue 2, 2000, Pages 203-227

  Mikami, T ^a  

^a HOKKAIDO UNIVERSITY   (Japan)

Author keywords

[No Author keywords available]

Indexed keywords

DYNAMICS; ENTROPY; GIBBS FREE ENERGY; INTEGRAL EQUATIONS; MARKOV PROCESSES; PROBLEM SOLVING; VARIATIONAL TECHNIQUES;

FOKKER-PLANCK EQUATION; GIBBS-BOLTZMANN ENTROPY FUNCTIONAL; NELSON STOCHASTIC MECHANICS; STOCHASTIC INTEGRAL EQUATION; VARIATIONAL FORMULATION; WASSERTEIN METRIC;

ASYMPTOTIC STABILITY;

EID: 0034262838     PISSN: 00954616     EISSN: None     Source Type: Journal    
DOI: 10.1007/s002450010008     Document Type: Article

References (27)

1. Breiman L (1992) Probability. Classics in Applied Mathematics, Vol 7. SIAM, Philadelphia, PA
2. Carlen EA (1984) Conservative diffusions. Comm Math Phys 94:293-315
3. Carmona R (1987) Probabilistic construction of Nelson processes, in Probabilistic Methods in Mathematical Physics, Itô K, Ikeda N, eds, pp 55-81. Kinokuniya, Tokyo
4. Cuesta JA, Tuero-Diaz A (1993) A characterization for the solution of the Monge-Kantorovich mass transference problem. Statist Probab Lett 16:147-152
5. Evans LC (1997) Partial differential equations and Monge-Kantorovich mass transfer. Preprint
6. Fleming WH, Soner HM (1993) Controlled Markov Processes and Viscosity Solutions. Springer-Verlag, Berlin
7. Föllmer H (1986) Time reversal on Wiener space, in Stochastic Processes - Mathematics and Physics, Albeverio S, Blanchard Ph, Streit L, eds, pp 119-129. Lecture Notes in Mathematics, Vol 1158. Springer-Verlag, Berlin
8. Freidlin MI, Wentzell AD (1984) Random Perturbations of Dynamical Systems. Springer-Verlag, Berlin
9. Friedman A (1975) Stochastic Differential Equations and Applications, I. Academic Press, New York
10. Gangbo W, McCann RJ (1996) The geometry of optimal transportation. Acta Math 177:113-161
11. Hartman P (1964) Ordinary Differential Equations. Wiley, New York
12. Jordan R, Kinderlehrer D, Otto F (1998) The variational formulation of the Fokker-Planck equation. SIAM J Math Anal 29:1-17
13. Liptser RS, Shiryaev AN (1977) Statistics of Random Processes, I, General Theory. Springer-Verlag, Berlin
14. Mikami T (1990) Variational processes from the weak forward equation. Commun Math Phys 135:19-40
15. Nagasawa M (1964) Time reversion of Markov processes. Nagoya Math J 24:117-204
16. Nagasawa M (1989) Transformations of diffusion and Schrödinger process. Probab Theory Rel Fields 82:109-136
17. Nelsen RB (1999) An Introduction to Copulas. Lecture Notes in Statistics, Vol 139. Springer-Verlag, Berlin
18. Nelson E (1967) Dynamical Theories of Brownian Motion. Princeton University Press, Princeton, NJ
19. Nelson E (1984) Quantum Fluctuations. Princeton University Press, Princeton, NJ
20. Otto F (1998) Dynamics of labyrinthine pattern formation in magnetic fluids: a mean-field theory. Arch Rational Mech Anal 141:63-103
21. Quastel J, Varadhan SRS (1997) Diffusion semigroups and diffusion processes corresponding to degenerate divergence form operators. Comm Pure Appl Math 50:667-706
22. Rachev ST, Rüschendorf L (1999) Mass Transportation Problems, Vol I. Springer-Verlag, Berlin
23. Schweizer B (1991) Thirty years of copulas, in Advances in Probability Distributions with Given Marginals, Dall'Aglio G, Kotz S, Salinetti G, eds, pp 13-50. Mathematics and its Applications, Vol 67. Kluwer, Dordrecht
24. Schweizer B, Sklar A (1983) Probabilistic Metric Space. North-Holland, New York
25. Sheu SJ (1991) Some estimates of the transition density of a nondegenerate diffusion Markov process. Ann Probab 19:538-561.
26. Stroock DW, Varadhan SRS (1979) Multidimensional Diffusion Processes. Springer-Verlag, Berlin
27. Zheng WA (1985) Tightness results for laws of diffusion processes application to stochastic mechanics. Ann Inst H Poincaré 21:103-124.