Rate of convergence for the Smoluchowski–Kramers approximation for distribution-dependent SDEs driven by fractional Brownian motions

In this paper, we study the rate in the Smoluchowski–Kramers approximation for the solution of the following distribution-dependent SDE driven by fractional Brownian motion [Formula: see text] where [Formula: see text] denotes the law of [Formula: see text], [Formula: see text] is a [Formula: see text]-dimensional fractional Brownian motion with Hurst parameter [Formula: see text]. Based on the techniques of multiple integrals and Malliavin calculus, we provide an explicit bound on total variation distance for the rate of convergence.