Pathwise estimates for an effective dynamics

TL;DR

This paper establishes pathwise error bounds for a scalar Markov process approximating the first component of overdamped Langevin dynamics, providing a rigorous quantitative measure of the approximation's accuracy over time.

Contribution

It introduces a method to bound the trajectorial error between the true dynamics and the approximation, extending previous marginal-based results to pathwise estimates.

Findings

Provides an upper bound on the supremum of the error over time.

Demonstrates the technique's applicability to quantitative averaging results.

Extends previous work from marginal to trajectorial error analysis.

Abstract

Starting from the overdamped Langevin dynamics in $\mathbb{R}^n$, $$ dX_t = -\nabla V(X_t) dt + \sqrt{2 \beta^{-1}} dW_t, $$ we consider a scalar Markov process $\xi_t$ which approximates the dynamics of the first component $X^1_t$. In the previous work [F. Legoll, T. Lelievre, Nonlinearity 2010], the fact that $(\xi_t)_{t \ge 0}$ is a good approximation of $(X^1_t)_{t \ge 0}$ 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 $\mathbb{E} \left( \sup_{0 \leq t \leq T} \left| X^1_t - \xi_t \right| \right)$, 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.