We consider the Lévy flow of diffeomorphisms of a manifold obtained by solving a stochastic differential equation driven by an n-dimensional Lévy process and n complete smooth vector fields (which generate a finite-dimensional Lie algebra L) and show that a necessary and sufficient condition for a large class of such flows to have an invariant measure is that each of the vector fields is divergence-free. We investigate the existence and uniqueness of such measures and examine their invariance with respect to the action of the associated Markov semigroup. In particular, we prove that there is an invariant probability measure on M if and only if the transformation Lie group G whose Lie algebra is L is compact. In the unbounded case we show that if there is a unique G-invariant measure, then it is also the unique invariant measure for the Markov semigroup provided the flow is recurrent.

There are extensive works on the limit theorems for sequences of stochastic ordinary differential equations written in the form:

where is a stochastic process and is a deterministic function, both of which take values in the space of vector fields. The case where {ftn} n satisfies certain mixing conditions has been studied by Khas’minskii [7], Kesten-Papanicolaou [6] and others.

Let H be a separable Hilbert space with inner product (,) and norm ║ ║. We denote by K the set of all linear operators on H. Let be a probability space and suppose we are given a family of σ-fields t≥O such that for O ≤ s ≤ t and .

Let (x, ζ,ℬt,Px) be a (standard) Markov process with state space 5 defined on the abstract space Ω. Here, xt is the sample path, ζ is the terminal time and ℬt is the smallest α-field of Ω in which xs s ≤ t are measurable. Let P’x, x ∈ S be another family of Markovian measures defined on (ℬt, Ω).

Theory of real and time continuous martingales has been developed recently by P. Meyer [8, 9]. Let be a square integrable martingale on a probability space P. He showed that there exists an increasing process ‹X›t such that