A multiflow approximation to diffusions

In this paper we introduce a new form of approximation to diffusions represented as solutions to stochastic differential equations. The approximants are generated by products of the flows of the vector fields defining the stochastic differential equation. For each interval of a given partition, each flow is followed for a period of time determined by the increment of a certain scalar random process over the interval. It is proved that under certain smoothness conditions the approximants converge to the diffusion as the mesh-size of the partitions goes to zero. The convergence is uniform in probability and the rate of convergence is calculated as a function of the mesh-size. For time independent vector fields on a manifold and equi-spaced partitions the approximants are considered as homogeneous Markov chains, called the approximating Markov chains. It is shown that any weakly convergent sub-sequence of invariant probability measures of a sequence of approximating Markov chains converges to an invariant probability measure of the diffusion.