Ergodic properties of markov processes driven by a set of vector fields
- 1 August 1989
- journal article
- research article
- Published by Taylor & Francis in Stochastics and Stochastic Reports
- Vol. 27 (4) , 235-247
- https://doi.org/10.1080/17442508908833577
Abstract
In this paper we describe the decomposition of a connected paracompact manifold M into uniquely ergodic components of a Markov process driven by a set of vector fields. Let Xr, O≤r ≤p be smooth and complete vector fields on M, let ξr(t) denote the 1-parameter groups generated by the vectors Xr. Given p+1 independent random variables χ0oo,χp we define a homogeneous Markov chain Zk, k= 1,2,..., with the following transition probabilities P(m,A)= P{[d]} Assuming the variables χr: have positive continuous densities on a small interval around the origin we prove that Zk is uniquely ergodic on each orbit of the group of diffeomorphisms generated by the flows ξr(t). Furthermore we point to the possible connections between the above result and the study of the ergodic decomposition of degenerate diffusions on manifolds.Keywords
This publication has 5 references indexed in Scilit:
- On unique ergodicity for degenerate diffusionsStochastics, 1987
- Flows of stochastic dynamical systems: ergodic theoryStochastics, 1985
- Stochastic differential equations and stochastic flows of diffeomorphismsPublished by Springer Nature ,1984
- Orbits of families of vector fields and integrability of distributionsTransactions of the American Mathematical Society, 1973
- Functional AnalysisPublished by Springer Nature ,1968