Uniform-in-Time Smoluchowski-Kramers Approximation in Total Variation for Fractional SDEs

Abstract

Let $B^H$ be a one-dimensional fractional Brownian motion with Hurst index $H\in(1/2,1)$. We study the small-mass limit of the kinetic equation \[ dX_t^\mu=Y_t^\mu\,dt,\qquad \mu\,dY_t^\mu=b(X_t^\mu)\,dt-Y_t^\mu\,dt +\sigma(X_t^\mu)\,dB_t^H, \] where the noise coefficient is state dependent and uniformly nondegenerate. The limiting equation is the Young differential equation \[ dX_t=b(X_t)\,dt+\sigma(X_t)\,dB_t^H. \] Under uniform ellipticity and strict dissipativity of the transformed drift, we prove that, for every $t_0>0$ and every $\rho<2H-1$, \[ \sup_{t\ge t_0}d_{TV}\big(X_t^\mu,X_t\big) \le C_{t_0,\rho}\mu^\rho . \] The estimate is uniform on the entire half-line. We also construct stationary solutions on a two-sided fractional-noise space and obtain convergence of their one-time marginals in total variation. The exponent $2H-1$ is identified as the natural endpoint: it is generated by the quadratic velocity term after the Lamperti transform, and it degenerates as $H\downarrow1/2$. A nonconstant uniformly elliptic example is included.