A multiflow approximation to diffusions (Q811018)

The author introduces a new form of approximation to diffusions represented as solutions of stochastic differential equations. In the various forms of approximations, one solves a sequence of ordinary differential equations which are derived by substituting, in place of Brownian path, a piecewise differentiable approximation corresponding to a given partition. Then for a given fixed partition, one must solve a new set of differential equations for each Brownian path. To overcome this difficulty, author introduces the so-called multiflow approximation. Here, the approximants are generated by products of the flows of the vector fields defining the stochastic differential equation. Under certain smoothness conditions, it is proved that the approximants converge to a diffusion as the mesh-size of the partitions go to zero. For time independent vector fields on a manifold, the approximants are considered as homogeneous Markov chains, and it is shown that the invariant probability measures of a sequence of approximating Markov chains converge weakly to an invariant probability measure of a diffusion.