Ergodic properties of markov processes driven by a set of vector fields

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 X_r, O≤r ≤p be smooth and complete vector fields on M, let ξ_r(t) denote the 1-parameter groups generated by the vectors X_r. Given p+1 independent random variables χ₀oo,χ_p we define a homogeneous Markov chain Z_k, k= 1,2,..., with the following transition probabilities P(m,A)= P{[d]} Assuming the variables χ_r: have positive continuous densities on a small interval around the origin we prove that Z_k is uniquely ergodic on each orbit of the group of diffeomorphisms generated by the flows ξ_r(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:

• Markov chain
• ergodic decomposition
• invariant measures
• approximation of stochastic differential equations

^*Present address: Division of Applied Mathematices, Brown University, Providence, RI 02912, USA

^*Present address: Division of Applied Mathematices, Brown University, Providence, RI 02912, USA

Notes

^*Present address: Division of Applied Mathematices, Brown University, Providence, RI 02912, USA