Existence and uniqueness of solutions to Fokker-Planck type equations with irregular coefficients

Communications in Partial Differential Equations

Volumn 33, Issue 7, 2008, Pages 1272-1317

  Le Bris, C ^a,b   Lions, P L ^c,d  

^a UNIVERSITÉ PARIS EST   (France)

^b INRIA ROCQUENCOURT   (France)

^c l'alimentation et l'Environnement   (France)

^d UNIVERSITÉ PARIS DAUPHINE   (France)

Author keywords

DiPerna Lions theory; Fokker Planck equations; Kolmogorov equations; Linear transport equations; Renormalized solutions

Indexed keywords

EID: 46349102591     PISSN: 03605302     EISSN: 15324133     Source Type: Journal    
DOI: 10.1080/03605300801970952     Document Type: Article

References (24)

1. Ambrosio, L. (2004). Transport equation and Cauchy problem for BV vector fields. Inventiones Mathematicae 158:227-260.
2. Ambrosio, L., Lecumberry, M., Maniglia, S. (2005). Lipschitz regularity and approximate differentiability of the DiPerna-Lions flow. Rendiconti del Seminario Fisico Matematico di Padova 114:29-50.
3. Ambrosio, L., Crippa, G. (2008). Existence, uniqueness, stability and differentiability properties of the flow associated to weakly differentiable vector fields. In: Ancona, F. et al., eds., Transport Equations and Multi-D Hyperbolic Conservation Laws. Berlin: Springer. Lecture Notes of the Unione Matematica Italiana 5, pp. 3-57.
4. Barles, G., Lions, P.-L. (1995). Remarques sur les problèmes de réflexion oblique [Remarks on oblique reflection problems]. C. R. Acad. Sc. 320, Série 1:69-74.
5. Di Perna, R. J., Lions, P. L. (1989). Ordinary differential equations, transport theory and Sobolev spaces. Invent. Math. 98(3):511-547.
6. Doss, H. (1977). Liens entre équations différentielles stochastiques et ordinaires [Relation between stochastic and ordinary differential equations]. Ann. IHP Sec. B 13(2):99-125.
7. Doss, H. (1976). Liens entre équations différentielles stochastiques et ordinaires [Relation between stochastic and ordinary differential equations]. C. R. Acad. Sc. Paris, Sér. A-B 283(13):A939-A942.
8. Figalli, A. (2008). Existence and uniqueness of martingale solutions for SDEs with rough or degenerate coefficients. J. Funct. Anal. 254:109-153.
9. Fournié, E., Lasry, J.-M., Lebuchoux, J., Lions, P.-L. (2001). Applications of Malliavin calculus to Monte-Carlo methods in finance. II. Finance Stoch. 5(2):201-236.
10. Hauray, M., Le Bris, C., Lions, P. L. (2007). Deux remarques sur les flots généralisés d'équations différentielles ordinaires [Two remarks on generalized flows for ordinary differential equations]. C. R. Acad. Sc., Paris, Sér. I., Math. 344, Série 1 759-764.
11. Ikeda, N., Watanabe, S. (1989). Stochastic Differential Equations and Diffusions Processes. 2nd ed. North-Holland.
12. Karatzas, I., Shreve, S. (1988). Brownian Motion and Stochastic Calculus. Graduate Texts in Mathematics, Vol. 113. Springer.
13. Lerner, N. (2004). Transport equations with partially BV velocities. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3(4):681-703.
14. 1,1 velocities and applications. Annali di Matematica Pura ed Applicata 183:97-130.
15. Le Bris, C., Lions, P. L. Generalized flows for stochastic differential equations with irregular coefficients. In preparation.
16. Lions, P. L. (1998). Sur les équations différentielles ordinaires et les équations de transport [On ordinary differential equations and transport equations]. C. R. Acad. Sc. Paris, Sér. I., Math. 326(7):833-838.
17. Osada, H. (1987). Diffusion processes with generators of generalized divergence form. J. Math. Kyoto Univ. 27(4):597-619.
18. Ottinger, H.-C. (1996). Stochastic Processes in Polymeric Fluids. Springer.
19. Owens, R. G., Phillips, T. N. (2002). Computational Rheology. Imperial College Press.
20. Revuz, D., Yor, M. (1999). Continuous Martingales and Brownian Motion. 3rd ed. Grundlehren der mathematischen Wissenschaften, 293. Springer.
21. Rogers, L. C. G., Williams, D. (2000). Diffusions, Markov Processes and Martingales. 2nd ed. Cambridge Mathematical Library, 2 volumes.
22. Stroock, D., Varadhan, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der mathematischen Wissenschaften, 233. Springer.
23. Sussmann, H. J. (1978). On the gap between deterministic and stochastic differential equations. Ann. Probability 6(1):19-41.
24. Sussmann, H. J. (1977). An interpretation of stochastic differential equations as ordinary differential equations which depend on the sample point. Bull. A. M. S. 83(2):296-298.