On stability of velocity vectors for some passive tracer models. (English) Zbl 1204.60070
In the first part of this paper, the authors extends the sufficient conditions for the existence of an invariant measure for Markov semigroups from T. Komorowski, S. Peszat and T. Szarek [Ann. Probab. 38, No. 4, 1401–1443 (2010; Zbl 1214.60035)]. They also formulate criteria for stability and prove results concerning the sweeping property of a Markov semigroup. In the second part, applications are given to general stochastic differential equations. In particular, the general results apply to the equation describing the passive tracer in a compressible random flow. See A. Fannjiang, T. Komorowski and S. Peszat [Stochastic Processes Appl. 97, No. 2, 171–198 (2002; Zbl 1058.60048)]. It is found that the velocity of the passive tracer converges weakly to some random vector.
Reviewer: Marius Iosifescu (Bucureşti)
MSC:
60J25 | Continuous-time Markov processes on general state spaces |
60H15 | Stochastic partial differential equations (aspects of stochastic analysis) |
76N10 | Existence, uniqueness, and regularity theory for compressible fluids and gas dynamics |