A new topological approach to the L∞-uniqueness of operators and the L1-uniqueness of Fokker-Planck equations

Journal of Functional Analysis

Volumn 241, Issue 2, 2006, Pages 557-610

  Wu, Liming ^a,b   Zhang, Yiping ^a  

^a WUHAN UNIVERSITY   (China)

^b UNIVERSITÉ BLAISE PASCAL   (France)

Author keywords

Diffusions; Fokker Planck equations; Liouville property; Schr dinger operators; Uniqueness of operators in L

Indexed keywords

EID: 33749997456     PISSN: 00221236     EISSN: 10960783     Source Type: Journal    
DOI: 10.1016/j.jfa.2006.04.020     Document Type: Article

References (51)

1. Aizenman M., and Simon B. Brownian motion and Harnack inequality for Schrödinger operators. Comm. Pure Appl. Math. 35 2 (1982) 210-273
2. Arendt W. The abstract Cauchy problem, special semigroups and perturbation. In: Nagel R. (Ed). One Parameter Semigroups of Positive Operators. Lecture Notes in Math. vol. 1184 (1986), Springer, Berlin
3. Azencott R. Behavior of diffusion semi-groups. Bull. Soc. Math. France 102 (1974) 193-240
4. Bakry D. L'hypercontractivité et son utilisation en théorie des semigroupes. École d'Eté de Probabilités de Saint-Flour, 1992. Lecture Notes in Math. vol. 1581 (1994), Springer 1-114
5. Berthier A.M., and Gaveau B. Critères de convergence des fonctionnelles de Kac et application en mecanique quantique. J. Funct. Anal. 29 (1978) 416-424
6. Bogachev V.I., and Röckner M. Invariant measures of diffusion processes: Regularity, existence, and uniqueness problems. Stochastic Partial Differential Equations and Applications. Trento, 2002. Lecture Notes Pure Appl. Math. vol. 227 (2002), Dekker, New York 69-87
7. Cerrai S. Second Order PDE's in Finite and Infinite Dimension. A Probabilistic Approach. Lecture Notes in Math. vol. 1762 (2001), Springer, Berlin
8. Chung K.L., and Zhao Z.X. From Brownian Motion to Schrödinger's Equations. Grundlehren Math. Wiss. vol. 312 (1995), Springer
9. Cycon H.L., Froese R.G., Kirsch W., and Simon B. Schrödinger Operators, with Applications to Quantum Mechanics and Global Geometry. Texts Monogr. Phys. (1987), Springer, Berlin
10. Davies E.B. One-Parameter Semigroups (1980), Academic Press, New York
11. 1-properties of second order elliptic operators. Bull. London Math. Soc. 17 (1985) 417-436
12. Dellacherie C., and Meyer P.A. Probabilités et Potentiels, vols. I-III (1976), Hermann, Paris 1980; 1983
13. p-uniqueness for one-dimensional diffusions, Mémoire de D.E.A., Université Blaise Pascal, 1997
14. Eberle A. Uniqueness and Non-Uniqueness of Semigroups Generated by Singular Diffusion Operators. Lecture Notes in Math. vol. 1718 (1999), Springer, Berlin
15. Ethier S.N., and Kurtz T.S. Markov Processes: Characterization and Convergence (1986), Wiley
16. Feller W. The parabolic differential equations and the associated semigroups of transformations. Ann. of Math. (2) 55 3 (1952) 468-519
17. Feller W. Semi-groups of transformations in general weak topologies. Ann. of Math. (2) 57 (1953) 287-308
18. Grigor'yan A.A. On stochastically complete manifolds. Soviet Math. Dokl. 34 (1987) 310-313
19. Grigor'yan A.A. Stochastically complete manifolds and summable harmonic functions. Izv. AN SSSR, Ser. Math. 52 (1988) 1102-1108 (in Russian); English transl. in:
20. Grigor'yan A.A. Stochastically complete manifolds and summable harmonic functions. Math. USSR Izv. 33 (1989) 425-432
21. Ikeda N., and Watanabe S. Stochastic Differential Equations and Diffusion Processes (1981), North-Holland, Kodansha, Amsterdam, Tokyo
22. L. Karp, P. Li, The heat equation on complete Riemannian manifolds, preprint, Stanford and Ann Arbor, 1983 (unpublished)
23. Kato T. Schrödinger operators with singular potentials. Israel J. Math. 13 (1972) 135-148
24. Kato T. Perturbation Theory for Linear Operators. second ed. (1984), Springer, Berlin
25. 0-semigroup on a general locally convex vector space, preprint, 2006
26. 1 solution for the Laplace equation and the heat equation on Riemannian manifolds. J. Differential Geom. 20 (1984) 447-457
27. p and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math. 153 (1984) 279-301
28. Li X.D. Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl. 84 (2005) 1295-1361
29. Liskevitch V. On the uniqueness problem for Dirichlet operators. J. Funct. Anal. 162 (1999) 1-13
30. Liskevitch V., and Semenov Y. Dirichlet operators: A priori estimates and the uniqueness problem. J. Funct. Anal. 109 (1992) 199-213
31. Malliavin P. Asymptotic of Green Function's function on a Riemannian manifold and Ito's stochastic integral. Proc. Natl. Acad. Sci. USA 71 (1974) 381-383
32. Oksendal B. Stochastic Differential Equations. An Introduction with Applications. sixth ed. (2003), Springer, New York
33. Pazy A. Semigroups of Linear Operators and Applications to Partial Differential Equations. Appl. Math. Sci. vol. 44 (1983), Springer, Berlin
34. Reed M., and Simon B. Methods of Modern Mathematical Physics, II, Fourier Analysis, Self-Adjointness (1975), Academic Press, New York
35. Schafer H.H. Topological Vector Spaces. Grad. Texts in Math. third ed. vol. 3 (1971), Springer, Berlin
36. Schwartz L. Lectures on Mixed Problems in Partial Differential Equations and the Representation of Semigroups (1958), Tata Inst. Fund. Research
37. Simon B. Schrödinger semigroups. Bull. Amer. Math. Soc. (N.S.) 7 3 (1982) 447-526
38. 1: Existence, uniqueness and associated Markov processes. Ann. Sc. Norm. Sup. Pisa Cl. Sci. 28 (1999) 99-140
39. Strichartz R. Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52 (1983) 48-79
40. p-Liouville properties. J. Reine Angew. Math. 456 (1994) 173-196
41. Sturm K.T. Analysis on local Dirichlet spaces II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32 (1995) 275-312
42. Wielens N. On the essential self-adjointness of generalized Schrödinger operators. J. Funct. Anal. 61 (1985) 98-115
43. Wu L. Uniqueness of Schrödinger operators restricted to a domain. J. Funct. Anal. 153 (1998) 276-319
44. Wu L. Uniqueness of Nelson's diffusions. Probab. Theory Related Fields 114 (1999) 549-585
45. Wu L. Uniqueness of Nelson's diffusions (II): Infinite-dimensional setting and applications. Potential Anal. 13 (2000) 269-301
46. p-uniqueness of Schrödinger operators and the capacitary positive improving property. J. Funct. Anal. 182 (2001) 51-80
47. ∞, a new topological approach. C. R. Acad. Sci. Paris Sér. I 334 (2002) 699-704
48. Yau S.T. Harmonic functions on complete Riemannian manifolds. Comm. Pure Appl. Math. 28 (1975) 201-228
49. Yau S.T. Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry. Indiana Math. J. 25 (1976) 659-670
50. Yosida K. Functional Analysis. Grundlehren Math. Wiss. third ed. vol. 123 (1971), Springer
51. Ziemer W.P. Weakly Differentiable Functions (1989), Springer, Berlin