Diffusion approximation for transport equations with dissipative drifts

We study stochastic differential equations with a small perturbation
parameter. Under the dissipative condition on the drift coefficient and the local
Lipschitz condition on the drift and diffusion coefficients we prove the existence
and uniqueness result for the perturbed SDE, also the convergence result for the
solution of the perturbed system to the solution of the unperturbed system when
the perturbation parameter approaches zero. We consider the application of the
above-mentioned results to the Cauchy problem and the transport equations.