Entropy, reversible diffusion processes, and Markov uniqueness

Journal of Functional Analysis

Volumn 138, Issue 1, 1996, Pages 243-272

  Cattiaux, Patrick ^a,b   Fradon, Myriam ^c  

^a CENTRE DE MATHÉMATIQUES APPLIQUÉES   (France)

^b UNIVERSITÉ PARIS OUEST NANTERRE LA DÉFENSE   (France)

^c UNIVERSITÉ PARIS SACLAY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0030600948     PISSN: 00221236     EISSN: None     Source Type: Journal    
DOI: 10.1006/jfan.1996.0064     Document Type: Article

References (41)

1. [AHKS77] S. ALBEVERIO, R. HOEGH-KROHN, AND L. STREIT, Energy forms, Hamiltonians, and distorted Brownian paths, J. Math. Phys. 18, No. 1 (1977), 907-917.
2. [AKR90] S. ALBEVERIO, S. KUSUOKA, AND M. RÖCKNER, On partial integration in infinite dimensional spaces and application to Dirichlet forms, J. London Math. Soc. 42 (1990), 122-136.
3. [AM91] S. ALBEVERIO AND Z. MA, Necessary and sufficient conditions for the existence of m-perfect processes associated with Dirichlet forms, in "Séminaire de proba. XXV," Lecture Notes in Math., Vol. 1485, Springer-Verlag, New York/Berlin, 1991.
4. [AR89] S. ALBEVERIO AND M. RÖCKNER, Classical Dirichlet forms on topological vector spaces - The construction of the associated diffusion process, Probab. Theory Relat. Fields 83 (1989), 405-434.
5. [AR90] S. ALBEVERIO AND M. RÖCKNER, Classical Dirichlet forms on topological vector spaces - closability and Cameron-Martin formula, J. Funct. Anal. 88 (1990), 395-436.
6. [AR91] S. ALBEVERIO AND M. RÖCKNER, Stochastic differential equations in infinite dimensions: Solutions via Dirichlet forms, Probab. Theory Relat. Fields 89 (1991), 347-386.
7. [ARZ93a] S. ALBEVERIO, M. RÖCKNER, AND T. S. ZHANG, Girsanov transform for symmetric diffusions with infinite dimensional state space, Ann. Probab. 21, No. 2 (1993), 961-978.
8. [ARZ93b] S. ALBEVERIO, M. RÖCKNER, AND T. S. ZHANG, Markov uniqueness for a class of infinite dimensional Dirichlet operators, Stochast. Processes Optim. Control. Stochast. Monographs 7 (1993), 1-26.
9. [BG68] R. M. BLUMENTHAL AND R. K. GETOOR, "Markov Processes and Potential Theory," Academic Press, New York/London, 1968.
10. [BR94a] V. I. BOGACHEV AND M. RÖCKNER, Hypoellipticity of invariant measures of infinite dimensional diffusions, C.R. Acad. Sci. Paris, Sér. 1 318 (1994), 553-558.
11. [BR94b] V. I. BOGACHEV AND M. RÖCKNER, Regulatity of invariant measures on finite and infinite dimensional spaces and applications, preprint, 1994.
12. [Car84] E. CARLEN, Conservative diffusions, Comm. Math. Phys. 94 (1984), 293-316.
13. [Car85] R. CARMONA, Probabilistic construction of Nelson processes, in "Taniguchi Symposium PMMP Katata, 1985," pp. 55-81.
14. [CL94] P. CATTIAUX AND C. LÉONARD, Minimization of the Kullback information of diffusion processes, Ann. Inst. H. Poincaré 30, No. 1 (1994), 83-132; Correction, Ann. Inst. H. Poincaré (1995), to appear.
15. [CL94] P. CATTIAUX AND C. LÉONARD, Minimization of the Kullback information of diffusion processes, Ann. Inst. H. Poincaré 30, No. 1 (1994), 83-132; Correction, Ann. Inst. H. Poincaré (1995), to appear.
16. [CL95a] P. CATTIAUX AND C. LÉONARD, Large deviations and Nelson processes, Forum Math. 7 (1995), 95-115.
17. [CL95b] P. CATTIAUX AND C. LÉONARD, Minimization of the Kullback information for general markov processes, to appear in Séminaire de Proba. XXX.
18. [CP95] P. CATTIAUX AND F. PETIT, preprint, 1995.
19. [DM87] C. DELLACHERIE AND P. A. MEYER, "Probabilités et potentiel," Vol. 4, Hermann, Paris, 1987.
20. [Fit89] P. J. FITZSIMMONS, Markov processes and non-symmetric Dirichlet forms without regulatity, J. Funct. Anal. 85 (1989), 287-306.
21. [FÖ172] H. FÖLLMER, The exit measure of a supermartingale, Z. Warsch. 21 (1972), 154-166.
22. [FÖ184] H. FÖLLMER, Time reversal on Wiener space, in "Proceedings Bielefeld 84," Lecture Notes in Math., Vol. 1158, 119-129, Springer-Verlag, Berlin, 1984.
23. [FÖ188] H. FÖLLMER, Random fields and diffusion processes, in "École d'été de probabilités de Saint-Flour," Lecture Notes in Math., Vol. 1362, pp. 101-204, Springer-Verlag, New York/Berlin, 1988.
24. [FP94] H. FÖLLMER AND P. PROTTER, An extension of K. Ito's change of variables formula, preprint, 1994.
25. [Fra94] M. FRADON, Diffusions réfléchies réversibles dégénérées, to appear in Potential Analysis.
26. [Fra95a] M. FRADON, "Diffusions dégénérées, réfléchies ou à dérives singulières: étude des lois et des formes de dirichlet associées," Ph.D. thesis, 1995.
27. [Fra95b] M. FRADON, Entropy, reversible diffusion processes and Markov uniqueness: The case of a general diffusion matrix, preprint, 1995.
28. [FT84] M. FUKUSHIMA AND M. TAKEDA, A transformation of a symmetric Markov process and the Donsker-Varadhan theory, Osaka J. Math 21 (1984), 311-326.
29. [Fuk80] M. FUKUSHIMA, "Dirichlet Forms and Markov Processes," Kodansha/North-Holland, Amsterdam, 1980.
30. [Gro93] L. GROSS, "Logarithmic Sobolev Inequalities and Contractivity Properties of Semi-groups in Dirichlet Forms," Lecture Notes in Math., Vol. 1563, pp. 54-88, Springer-Verlag, Berlin/New York, 1993.
31. [Jac79] J. JACOD, "Problèmes de martingales," Lecture Notes in Math., Vol. 714, Springer-Verlag, Berlin/New York, 1979.
32. [MR92] Z. M. MA AND M. ROCKNER, "An Introduction to the Theory of (Non-Symmetric) Dirichlet Forms," Springer-Verlag, Berlin, 1992.
33. [MZ84] P. A. MEYER AND W. A. ZHENG, Tightness criteria for laws of semimartingales, Ann. Inst. H. Poincaré 20 (1984), 353-372.
34. [MZ85] P. A. MEYER AND W. A. ZHENG, Construction de processus de Nelson réversibles, in "Séminaire Probab. XIX," Lecture Notes in Math., Vol. 1123, pp. 12-26, Springer-Verlag, New York/Berlin, 1985.
35. [Nel88] E. NELSON, Stochastic mechanics and random fields, in "École d'été de probabilités de Saint-Flour," Lecture Notes in Math., Vol. 1362, pp. 429-450, Springer-Verlag, New York/Berlin, 1988.
36. [RZ92] M. RÖCKNER AND T. S. ZHANG, Uniqueness of generalized Schrödinger operators and applications, J. Funct. Anal. 105 (1992), 187-231.
37. [RZ94] M. RÖCKNER AND T. S. ZHANG, Uniqueness of generalized Schrödinger operators, J. Funct. Anal. 119 (1994), 455-467.
38. [Son94a] S. SONG, Markov uniqueness and essential self-adjointness of perturbed Ornstein-Uhlenbeck operators, preprint, 1994.
39. [Son94b] S. SONG, A study on Markovian maximality, change of probability and regularity, Potential Analysis 3, No. 4 (1994), 391-422.
40. [Tak92] M. TAKEDA, The maximum Markovian self-adjoint extension of generalized Schrödinger operators, J. Math. Soc. Japan 44 (1992), 113-130.
41. [Zhe85] W. A. ZHENG, Tightness results for law of diffusion processes, application to stochastic mechanisms, Ann. Inst. H. Poincaré 21 (1985), 103-124.