On ergodicity of some Markov processes

By Tomasz Komorowski, S. Peszat, Tomasz Szarek. 2008

110 citations recorded by the source corpus.

We formulate a criterion for the existence and uniqueness of an invariant measure for a Markov process taking values in a Polish phase space. In addition, weak- * ergodicity, that is, the weak convergence of the ergodic averages of the laws of the process starting from any initial distribution, is established. The principal assumptions are the existence of a lower bound for the ergodic averages of the transition probability function and its local uniform continuity. The latter is called the e-property. The general result is applied to solutions of some stochastic evolution equations in Hilbert spaces. As an example, we consider an evolution equation whose solution describes the Lagrangian observations of the velocity field in the passive tracer model. The weak- * mean ergodicity of the corresponding invariant measure is used to derive the law of large numbers for the trajectory of a tracer.

Subfields: Finance, Management Information Systems, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0deffb3b67cce543ac5275ccca7f542b7b4d94e4

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Finance

1. A. Kulik
2. Tomasz Komorowski
3. M. Scheutzow
4. Jian Wang
5. Dejun Luo
6. A. Walczuk

In Mathematical Physics

1. Tomasz Komorowski
2. M. Scheutzow
3. Jian Wang
4. Dejun Luo
5. A. Kulik
6. A. Walczuk

Citations in the corpus

• Refined basic couplings and Wasserstein-type distances for SDEs with Lévy noises (2016, result): “pb(Rd) into itself, where Lipb(Rd) denotes the set of bounded Lipschitz functions on Rd. Such property is useful in studying the existence of a unique invariant probability for Markov semigroups, see [12] and [29, Section 2.2]. 1.2 Strong ergodicity We are also interested in obtaining the exponentia”
• Generalized couplings and convergence of transition probabilities (2015, background): “In [13] and [2], uniqueness and asymptotic stability were shown for so-called e-processes (which we explain below) and a similar approach was used in [11] to prove asymptotic stability.”
• Central limit theorem for Markov processes with spectral gap in the Wasserstein metric (2011, background): “The Markov processes considered in the aforementioned papers satisfy either the asymptotic strong Feller property introduced in [19], or a somewhat weaker e-property (see [24]).”