×

zbMATH — the first resource for mathematics

Exponential mixing for some SPDEs with Lévy noise. (English) Zbl 1239.60060
The authors show how gradient estimates for transition semigroups can be used to establish exponential mixing for a class of Markov processes in inifinte dimensions. The analysis is concentrated on semilinear systems driven by cylindrical \(\alpha\)-stable noises for \(0 < \alpha < 2\). First for bounded nonlinearities the authors prove the system to be ergodic and strong mixing. Then they show exponential mixing in case where the nonlinearity, or its Lipschitz constant, are sufficiently small. The analysis is done in a Hilbert space setting. The generator \(A\) of the involved strongly continuous contraction semigroup \((e^{tA})_{t \geq 0}\) is assumed to be symmetric and stictly negative with compact inverse. In particular, \(e^{tA}\) has to be compact for all \(t > 0\).

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
47D07 Markov semigroups and applications to diffusion processes
60J75 Jump processes (MSC2010)
35R60 PDEs with randomness, stochastic partial differential equations
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] DOI: 10.1016/j.spa.2008.03.006 · Zbl 1168.60014 · doi:10.1016/j.spa.2008.03.006
[2] DOI: 10.1023/A:1008705820024 · Zbl 0978.60096 · doi:10.1023/A:1008705820024
[3] DOI: 10.1016/S0304-4149(97)00112-9 · Zbl 0934.60055 · doi:10.1016/S0304-4149(97)00112-9
[4] DOI: 10.1016/j.crma.2010.01.022 · Zbl 1205.60119 · doi:10.1016/j.crma.2010.01.022
[5] DOI: 10.1016/0893-9659(91)90121-B · Zbl 0748.35052 · doi:10.1016/0893-9659(91)90121-B
[6] DOI: 10.1080/17442508708833459 · Zbl 0622.60072 · doi:10.1080/17442508708833459
[7] DOI: 10.1017/CBO9780511666223 · doi:10.1017/CBO9780511666223
[8] DOI: 10.1017/CBO9780511662829 · Zbl 0849.60052 · doi:10.1017/CBO9780511662829
[9] DOI: 10.1006/jfan.1995.1076 · Zbl 0832.60069 · doi:10.1006/jfan.1995.1076
[10] DOI: 10.1016/j.spa.2008.02.007 · Zbl 1157.60063 · doi:10.1016/j.spa.2008.02.007
[11] DOI: 10.1214/08-AOP392 · Zbl 1173.37005 · doi:10.1214/08-AOP392
[12] DOI: 10.1007/BFb0089647 · doi:10.1007/BFb0089647
[13] Marinelli C., Electron. J. Probab. 15 pp 1529– · Zbl 1225.60108 · doi:10.1214/EJP.v15-818
[14] DOI: 10.1090/S0033-569X-08-01090-5 · Zbl 1153.60037 · doi:10.1090/S0033-569X-08-01090-5
[15] DOI: 10.1016/B978-1-4832-0022-4.50006-5 · doi:10.1016/B978-1-4832-0022-4.50006-5
[16] DOI: 10.1017/CBO9780511721373 · Zbl 1205.60122 · doi:10.1017/CBO9780511721373
[17] S. Peszat and J. Zabczyk, Stochastic Partial Differential Equations and Applications VII, Lect. Notes Pure Appl. Math 245 (Chapman and Hall/CRC, 2006) pp. 229–242.
[18] DOI: 10.1239/aap/1275055234 · Zbl 1191.91061 · doi:10.1239/aap/1275055234
[19] DOI: 10.1080/07362990902976629 · Zbl 1181.60150 · doi:10.1080/07362990902976629
[20] Xu L., Electron. J. Probab. 15 pp 1994– · doi:10.1214/EJP.v15-831
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.