Smoluchowski-Kramers Approximation for Stochastic Differential Equations driven by Fractional Brownian Motion

Abstract

In this paper, we discuss the validity of an approximation inspired by the Smoluchowski-Kramers approximation for a class of stochastic differential equations driven by fractional Brownian motion with additive noise. By rewriting such equations in the form of slow-fast systems and decomposing the fast component into three parts, we investigate the small mass limit of these equations and derive the corresponding convergence rates. Furthermore, under certain regularity conditions, we study the large and moderate deviation principles for a class of stochastic differential equations driven by fractional Brownian motion with small multiplicative noise via the weak convergence approach.