Positive densities of transition probabilities of diffusion processes

Theory of Probability and its Applications

Volumn 53, Issue 2, 2009, Pages 194-215

  Bogachev, V I ^a   Rockner M ^b,c   Shaposhnikov, S V ^a  

^a MOSCOW STATE UNIVERSITY   (Russian Federation)

^b BIELEFELD UNIVERSITY   (Germany)

^c PURDUE UNIVERSITY   (United States)

Author keywords

Density of transition probability; Infinite dimensional diffusion; Parabolic equation; Stationary distribution

Indexed keywords

EID: 67249131117     PISSN: 0040585X     EISSN: 10957219     Source Type: Journal    
DOI: 10.1137/S0040585X97983523     Document Type: Article

References (24)

1. V. I. BOGACHEV, N. V. KRYLOV, AND M. RÖCKNER, On regularity of transition probabilities and invariant measures of singular diffusions under minimal conditions, Comm. Partial Differential Equations, 26 (2001), pp. 2037-2080.
2. S. V. SHAPOSHNIKOV, Positiveness of invariant measures of diffusion processes, Dokl. Math., 76 (2007), pp. 174-179.
3. G. METAFUNE, D. PALLARA, AND A. RHANDI, Global properties of invariant measures, J. Funct. Anal., 223 (2005), pp. 396-424.
4. V. I. BOGACHEV, M. RÖCKNER, AND S. V. SHAPOSHNIKOV, Global regularity and bounds for solutions of parabolic equations for probability measures, Theory Probab. Appl., 50 (2006), pp. 561-581.
5. V. I. BOGACHEV, N. V. KRYLOV, aNDM. RÖCKNER, Elliptic equations for measures: Regularity and global bounds of densities, J. Math. Pures Appl., 85 (2006), pp. 743-757.
6. V. I. BOGACHEV, M. RÖCKNER, AND S. V. SHAPOSHNIKOV, Estimates of densities of stationary distributions and transition probabilities of diffusion processes, Theory Probab. Appl., 52 (2008), pp. 209-236.
7. V. I. BOGACHEV AND M. RÖCKNER, A generalization of Khasminskii's theorem on the existence of invariant measures for locally integrable drifts, Theory Probab. Appl., 45 (2001), pp. 363-378.
8. V. I. BOGACHEV, G. DA PRATO, AND M. RÖCKNER, On parabolic equations for measures, Comm. Partial Differential Equations, 33 (2008), pp. 397-418.
9. V. I. BOGACHEV, M. RÖCKNER, AND W. STANNAT, Uniqueness of solutions of elliptic equations and uniqueness of invariant measures of diffusions, Sb. Math., 193 (2002), pp. 945-976.
10. V. I. BOGACHEV, G. DA PRATO, M. RÖCKNER, AND W. STANNAT, Uniqueness of solutions to weak parabolic equations for measures, Bull. Lond. Math. Soc., 39 (2007), pp. 631-640.
11. S. V. SHAPOSHNIKOV, The nonuniqueness of solutions to elliptic equations for probability measures, Dokl. Math., 77 (2008), pp. 401-403.
12. N. V. KRYLOV, Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces, J. Funct. Anal., 183 (2001), pp. 1-41.
13. N. V. KRYLOV, An analytic approach to SPDEs, in Stochastic Partial Differential Equations: Six Perspectives, R. Carmona and B. Rozovskii, eds., Amer. Math. Soc., RI, Providence, 1999, pp. 185-241.
14. N. S. TRUDINGER, Pointwise estimates and quasilinear parabolic equations, Comm. Pure Appl. Math., 21 (1968), pp. 205-226.
15. D. G. ARONSON AND J. SERRIN, Local behavior of solutions of quasilinear parabolic equations, Arch. Rational Mech. Anal., 25 (1967), pp. 81-122.
16. J. MOSER, A Harnack inequality for parabolic differential equations, Comm. Pure Appl. Math., 17 (1964), pp. 101-134.
17. M. SCHEUTZOW AND H. vON WEIZS̈ aCKER, Which moments of a logarithmic derivative imply quasiinvariance? Doc. Math., 3 (1998), pp. 261-272.
18. V. I. BOGACHEV AND M. RÖCKNER, Elliptic equations for measures on infinite-dimensional spaces and applications, Probab. Theory Relat. Fields, 120 (2001), pp. 445-496.
19. A. EBERLE, Uniqueness and Nonuniqueness of Semigroups Generated by Singular Diffusion Operators, Springer-Verlag, Berlin, 1999.
20. W. STANNAT, (Nonsymmetric) Dirichlet operators on L1: Existence, uniqueness and associated Markov processes, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 28 (1999), pp. 99-140.
21. W. STANNAT, Time-dependent diffusion operators on L1, J. Evol. Equ., 4 (2004), pp. 463-495.
22. V. I. BOGACHEV, Gaussian Measures, Amer. Math. Soc., Providence, RI, 1998.
23. A. AGRACHEV, S. KUKSIN, A. SARYCHEV, AND A. SHIRIKYAN, On finite-dimensional projections of distributions for solutions of randomly forced 2D Navier-Stokes equations, Ann. Inst. H. Poincaré, 43 (2007), pp. 399-415.
24. A. SHIRIKYAN, Qualitative properties of stationary measures for three-dimensional Navier- Stokes equations, J. Funct. Anal., 249 (2007), pp. 284-306.