Diffusion Approximation for Transport Equations with Dissipative Drifts

TL;DR

This paper investigates the behavior of stochastic differential equations with dissipative drifts under small perturbations, proving convergence to unperturbed solutions and applying results to transport equations.

Contribution

It establishes existence, uniqueness, and convergence results for perturbed SDEs with dissipative drifts, extending understanding of their limiting behavior.

Findings

Proved existence and uniqueness of solutions for perturbed SDEs.

Established convergence of solutions as perturbation parameter approaches zero.

Applied theoretical results to Cauchy and transport equations.

Abstract

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.