Uniqueness for solutions of Fokker-Planck equations on infinite dimensional spaces

Communications in Partial Differential Equations

Volumn 36, Issue 6, 2011, Pages 925-939

  Bogachev, Vladimir ^a   da Prato, Giuseppe ^b   Röckner, Michael ^c,d  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

^b SCUOLA NORMALE SUPERIORE   (Italy)

^c BIELEFELD UNIVERSITY   (Germany)

^d PURDUE UNIVERSITY   (United States)

Author keywords

Fokker Planck equations; Kolmogorov operators; Parabolic equations for measures; Stochastic PDEs

Indexed keywords

EID: 79551545976     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605302.2010.539657     Document Type: Article

References (25)

1. Ambrosio, L. (2004). Transport equation and Cauchy problem for BV vector fields. Invent. Math. 158:227-260.
2. Ambrosio, L. (2008). Transport equation and Cauchy problem for non-smooth vector fields. In: Dacorogna, B., Marcellini, P., eds. Calculus of Variations and Non-Linear Partial Differential Equations. Lecture Notes in Mathematics, Vol. 1927. New York: Springer, pp. 2-41.
3. Ambrosio, L., Figalli, A. (2009). On flows associated to Sobolev vector fields in Wiener spaces: An approach à la DiPerna-Lions. J. Funct. Anal. 256:179-214.
4. Bogachev, V., Da Prato, G., Röckner, M. (2004). Existence of solutions to weak parabolic equations for measures. Proc. London Math. Soc. 88:753-774.
5. Bogachev, V., Da Prato, G., Röckner, M. (2008). On parabolic equations for measures. Comm. Part. Diff. Eqs. 33:1-22.
6. Bogachev, V., Da Prato, G., Röckner, M. (2008). Parabolic equations for measures on infinite-dimensional spaces. Dokl. Math. 78:544-549.
7. Bogachev, V., Da Prato, G., Röckner, M. (2009). Fokker-Planck equations and maximal dissipativity for Kolmogorov operators with time dependent singular drifts in Hilbert spaces. J. Funct. Anal. 256:1269-1298.
8. Bogachev, V., Da Prato, G., Röckner, M. (2010). Existence results for Fokker-Planck equations in Hilbert spaces. In: Dalang, R., Dozzi, M., Russo, F., eds. Proceedings of the conference "Stochastic Analysis, Random Fields and Applications VI". Progress in Probability, Vol. 63. Boston: Birkhäuser, pp. 23-35.
9. Bogachev, V., Da Prato, G., Röckner, M. (2010). Existence and uniqueness of solutions for Fokker-Planck equations on Hilbert spaces. J. Evol. Equ. 10:487-509.
10. Bogachev, V., Da Prato, G., Röckner, M., Shaposhnikov, S.V. Nonlinear evolution equations for measures on infinite dimensional spaces. To appear in Proceedings of the conference "Stochastic Partial Differential Equations and Applications-VIII" Quaderni di Matematica. Naples: Dipartimento di Matematica Seconda Università di Napoli.
11. Bogachev, V., Da Prato, G., Röckner, M., Stannat, W. (2007). Uniqueness of solutions to weak parabolic equations for measures. Bull. Lond. Math. Soc. 39:631-640.
12. Bogachev, V.I., Röckner, M., Shaposhnikov, S.V. (2009). Nonlinear evolution and transport equations for measures. Dokl. Math. 80:785-789.
13. Bogachev, V.I., Röckner, M., Stannat, W. (1999). Uniqueness of invariant measures and essential m-dissipativity of diffusion operators on L1. In: Infinite Dimensional Stochastic Analysis (Amsterdam, 1999). Verh. Afd. Natuurkd. 1. Reeks. K. Ned. Akad. Wet., Vol. 52, Amsterdam: R. Neth. Acad. Arts Sci., Amsterdam, pp. 39-54.
14. Bogachev, V.I., Röckner, M., Stannat, W. (2002). Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions. Mat. Sb. 193:3-36 (in (Russian); English transl. in Sb. Math. 193:945-976.
15. Cerrai, S. (1995). Weakly continuous semigroups in the space of functions with polynomial growth. Dynam. Syst. Appl. 4:351-372.
16. Da Prato, G. (2004). Kolmogorov Equations For Stochastic PDEs. Boston: Birkhäuser.
17. Da Prato, G., Röckner, M., Wang, F. (2009). Singular stochastic equations on Hilbert spaces: Harnack inequalities for their transition semigroups. J. Funct. Anal. 257:992-1017.
18. Da Prato, G., Tubaro, L. (2001). Some results about dissipativity of Kolmogorov operators. Czechosl. Math. J. 51:685-699.
19. Da Prato, G., Zabczyk, J. (1992). Stochastic Equations in Infinite Dimensions. Cambridge: Cambridge University Press.
20. Di Perna, R.J., Lions, P.L. (1989). Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98:511-548.
21. Figalli, A. (2008). Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254:109-153.
22. Le Bris, C., Lions, P.L. (2008) Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients. Comm. Part. Diff. Eqs. 33:1272-1317.
23. Lorenz, T. Radon measures solving the Cauchy problem of the nonlinear transport equation. Available at http://www.ub-uniheidelberg.de/archiv/7252.
24. Maniglia, S. (2007). Probabilistic representation and uniqueness results for measure-valued solutions of transport equation. J. Math. Pures Appl. 87:601-626.
25. Priola, E. (1999). On a class of Markov type semigroups in spaces of uniformly continuous and bounded functions. Studia Math. 136:271-295.