The Rate of Convergence for the Smoluchowski-Kramers Approximation for Stochastic Differential Equations with FBM

Abstract

In this paper, we study the rate in the Smoluchowski–Kramers approximation for the solution of the equation $$X_t=x+B_t^H+\int _0^t b(X_s)ds$$ where $$\{B_t^H, t\in [0,T]\}$$ is a fractional Brownian motion with Hurst parameter $$H\in \big (\frac{1}{2},1\big )$$ . Based on the techniques of Malliavin calculus, we provide an explicit bound on total variation distance for the rate of convergence.