Smoluchowski--Kramers Approximation with State-Dependent Friction in Rough Path Topology
Qingming Zhao, Xueru Liu, Wei Wang
TL;DR
This paper extends the Smoluchowski–Kramers approximation to a setting with state-dependent damping within rough path topology. By recasting the second-order Langevin equation as a fast–slow system and constructing a rough-path lift, it proves convergence to a Stratonovich-corrected limit using averaging and Poisson-equation techniques. Central to the result are uniform moment bounds, an averaging principle for the fast component with Gaussian invariant measures, and a careful analysis of the level-2 rough-path lift, yielding convergence in $L^p$ of the rough-path metric. The findings generalize prior constant-friction results and provide a rigorous rough-path framework for SK limits with non-constant damping. This has implications for precise asymptotics of damped stochastic systems where friction depends on the state.
Abstract
Smoluchowski-Kramers approximation in rough path topology with state-dependent damping is explored. The second-order Langevin equation has a form of fast-slow system after suitable change-of-variable, and then its solution is lifted as a rough path in a natural manner. Moment estimates of both the original path and the lift are given, followed by which, averaging technique and convergence theorem in rough path topology are used to pass the limit.