Pathwise estimates for an effective dynamics

Starting from the overdamped Langevin dynamics in Rn, dXt=−∇V(Xt)dt+2β−1dWt, we consider a scalar Markov process ξt which approximates the dynamics of the first component Xt1. In the previous work (Legoll and Lelièvre, 2010), the fact that (ξt)t≥0 is a good approximation of (Xt1)t≥0 is proven in terms of time marginals, under assumptions quantifying the timescale separation between the first component and the other components of Xt. Here, we prove an upper bound on the trajectorial error E(sup0≤t≤T|Xt1−ξt|) 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.