Pathwise estimates for an effective dynamics

Abstract

Starting from the overdamped Langevin dynamics in ${\mathbb{R}}^n$, $$d{X}_t=-\nabla V(X_t)dt+\sqrt{2{\beta }^{-1}}d{W}_t,$$ we consider a scalar Markov process ${\xi }_t$ which approximates the dynamics of the first component $X_t^1$. In the previous work [F. Legoll, T. Lelievre, Nonlinearity 2010], the fact that $({\xi }_t{)}_{t\ge 0}$ is a good approximation of $(X_t^1{)}_{t\ge 0}$ is proven in terms of time marginals, under assumptions quantifying the timescale separation between the first component and the other components of $X_t$. Here, we prove an upper bound on the trajectorial error $\mathbb{E}\left({\sup }_{0\le t\le T}\left|X_t^1-{\xi }_t\right|\right)$, for any $T>0$, under a similar set of assumptions. We also show that the technique of proof can be used to obtain quantitative averaging results.