Lower estimates of transition densities and bounds on exponential ergodicity for stochastic PDE'S

Annals of Probability

Volumn 34, Issue 4, 2006, Pages 1451-1496

  Goldys, B ^a   Maslowski, B ^b,c  

^a UNIVERSITY OF NEW SOUTH WALES   (Australia)

^b INSTITUTE OF MATHEMATICS   (Czech Republic)

^c MATHEMATICAL INSTITUTE   (Czech Republic)

Author keywords

Density estimates; Ornstein Uhlenbeck bridge; Spectral gap; Stochastic semilinear system; Uniform exponential ergodicity; V ergodicity

Indexed keywords

EID: 33750192938     PISSN: 00911798     EISSN: None     Source Type: Journal    
DOI: 10.1214/009117905000000800     Document Type: Article

References (43)

1. AIDA, S. (1998). Uniform positivity improving property, Sobolev inequalities, and spectral gaps. J. Funct. Anal. 158 152-185. MR1641566
2. AIDA, S. and SHIGEKAWA, I. (1994). Logarithmic Sobolev inequalities and spectral gaps: Perturbation theory. J. Funct. Anal. 126 448-475. MR1305076
3. 2-semigroups and extremality of Gibbs states. J. Funct. Anal. 144 394-423. MR1432591
4. ARIMA, R. ( 1964). On general boundary value problem for parabolic equations. J. Math. Kyoto Univ. 4 207-243. MRO197997
5. CERRAI, S. (1999). Ergodicity for stochastic reaction-diffusion systems with polynomial coefficients. Stochastics Stochastics Rep. 67 17-51. MR1717811
6. CERRAI, S. (2001). Second Order PDE's in Finite and Infinite Dimensions. A Probabilistic Approach. Springer, Berlin. MR1840644
7. CHOJNOWSKA-MlCHALIK, A. (2001). Transition Semigroups for Stochastic Semilinear Equations on Hubert Spaces. Dissertationes Math. 396. MR1841090
8. CHOJNOWSKA-MICHALIK, A. and GOLDYS, B. (2002). Symmetric Ornstein-Uhlenbeck semigroups and their generators. Probab. Theory Related Fields 124 459-486. MR1942319
9. DA PRATO, G. and ZABCZYK, J. (1992). Stochastic Equations in Inifnite Dimensions. Cambridge Univ. Press. MR1207136
10. DA PRATO, G. and ZABCZYK, J. (1996). Ergodicity for Infinite Dimensional Systems. Cambridge Univ. Press. MR1417491
11. DA PRATO, G., DEBUSSCHE, A. and GOLDYS, B. (2002). Some properties of invariant measures of non symmetric dissipative stochastic systems. Probab. Theory Related Fields 123 355-380. MR1918538
12. DA PRATO, G. and ZABCZYK, J. (1992). Nonexplosion, boundedness, and ergodicity for stochastic semilinear equations. J. Differential Equations 98 181-195. MR1168978
13. 1-properties of intrinsic Schrödinger operators. J. Fund. Anal. 65 126-146. MR0819177
14. GOLDYS, B. and MASLOWSKI, B. (2001). Uniform exponential ergodicity of stochastic dissipative systems. Czechoslovak Math. J. 51 745-762. MR1864040
15. GOLDYS, B. and MASLOWSKI, B. (2005). On the Ornstein-Uhlenbeck bridge. Preprint.
16. GOLDYS, B. and MASLOWSKI, B. (2004). Exponential ergodicity for stochastic reaction-diffusion equations. In Stochastic Partial Differential Equations and Applications VII. Lecture Notes Pure Appl. Math. 245 115-131. Chapman Hall/CRC Press, Boca Raton, FL. MR2227225
17. GOLDYS, B. and MASLOWSKI, B. (2006). Exponential ergodicity for stochastic Burgers and 2D Navier-Stokes equations. J. Funct. Anal. 226 230-255. MR2158741
18. GONG, F., RÖCKNER, M. and Wu, L. (2001). Poincaré inequality for weighted first order Sobolev spaces on loop spaces. J. Funct. Anal. 185 527-563. MR1856276
19. HAIRER, M. (2002). Exponential mixing properties of stochastic PDEs through asymptotic coupling. Probab. Theory Related Fields 124 345-380. MR1939651
20. HAIRER, M. (2002). Exponential mixing for a PDE driven by a degenerate noise. Nonlinearity 15271-279. MR1888852
21. HASMINSKII, R. Z. (1980). Stochastic Stability of Differential Equations. Sijthoff & Noordhoff, Alphen aan den Rijn. MR0600653
22. p. Osaka J. Math. 37 603-624. MR1789439
23. LYONS, T. J. and ZHENG, W. A. (1990). On conditional diffusion processes. Proc. Roy. Soc. Edinburgh USA 243-255. MR1069520
24. JACQUOT, S. and ROYER, G. (1995). Ergodicite d'une classe d'equations aux derivees partielles stochastiques. C. R. Acad. Sci. Paris Ser. 1 Math. 320 231-236. MR1320362
25. MASIERO, F. (2005). Semilinear Kolmogorov equations and applications to stochastic optimal control. Appl Math. Optim. 51 201-250. MR2117233
26. MASLOWSKI, B. (1989). Strong Feller property for semilinear stochastic evolution equations and applications. In Stochastic Systems and Optimization. Lecture Notes in Control Inform. Sci. 136 210-224. Springer, Berlin. MR1180781
27. MASLOWSKI, B. and SEIDLER, J. (2000). Probabilistic approach to the strong Feller property. Probab. Theory Related Fields 118 187-210. MR1790081
28. MASLOWSKI, B. and SIMÃO, I. (1997). Asymptotic properties of stochastic semilinear equations by the method of lower measures. Colloq. Math. 72 147-171. MR1425551
29. MASLOWSKI, B. and SIMÃO, I. (2001). Long time behaviour of non-autonomous SPDE's. Stochastic Process. Appl. 95 285-309. MR1854029
30. MATTINGLY, J. C., STUART, A. M. and HIGHAM, D. J. (2002). Ergodicity for SDEs and approximations: Locally Lipschitz vector fields and degenerate noise. Stochastic Process. Appl. 101 185-232. MR1931266
31. MATTINGLY, J. C. (2002). Exponential convergence for the stochastically forced Navier-Stokes equations and other partially dissipative dynamics. Comm. Math. Phys. 230 421-462. MR1937652
32. MEYN, S. P. and TWEEDIE, R. L. (1993). Markov Chains and Stochastic Stability. Springer, London. MR1287609
33. MEYN, S. P. and TWEEDIE, R. L. (1994). Computable bounds for geometric convergence rates of Markov chains. Ann. Appl. Probab. 4981-1011. MR1304770
34. VAN NEERVEN, J. M. A. M. (1996). The Asymptotic Behaviour of Semigroups of Linear Operators. Birkhäuser, Basel. MR 1409370
35. VAN NEERVEN, J. M. A. M. (1998). Nonsymmetric Ornstein-Uhlenbeck semigroups in Banach spaces. J. Funct. Anal. 155 495-535. MR1624573
36. ROBERTS, G. and ROSENTHAL, J. (1997). Geometric ergodicity and hybrid Markov chains. Electron. Comm. Probab. 2 13-25. MR1448322
37. SEIDLER, J. (1997). Ergodic behaviour of stochastic parabolic equations. Czechoslovak Math. J. 47 277-316. MR1452421
38. SEIDMAN, T. I. (1988). How violent are fast controls. Math. Control Signals Systems 1 89-95. MR0923278
39. SHARDLOW, T. (1999). Geometric ergodicity for stochastic PDEs. Stochastic Anal. Appl. 17 857-869. MR1714903
40. STETTNER, L. (1994). Remarks on ergodic conditions for Markov process on Polish spaces. Bull. Polish Acad. Sci. Math. 42 103-114. MR1810695
41. SIMÃO, I. (1996). Pinned Ornstein-Uhlenbeck processes on an infinite-dimensional space. In Stochastic Analysis and Applications (Powys, 1995) 401-407. World Sci. Publishing, River Edge, NJ. MR 1453146
42. SOWERS, R. (1992). Large deviations for the invariant measure of a reaction-diffusion equation with non-Gaussian perturbations. Probab. Theory Related Fields 92 393-421. MR1165518
43. Wu, L. (2004). Essential spectral radius for Markov semigroups. I. Discrete time case. Probab. Theory Related Fields 128 255-321. MR2031227