  We study stochastic differential equations(SDEs) 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.
