Pathwise estimates for an effective dynamics

Starting from the overdamped Langevin dynamics in R-n, dX(t) = -del V(X-t)dt + root 2 beta(-1)dW(t), we consider a scalar Markov process xi(t) which approximates the dynamics of the first component X-t(1). In the previous work (Legoll and Lelievre, 2010), the fact that (xi(t))(t >= 0) is a good approximation of (X-t(1))(t >= 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 E (sup(0 <= t <= T) |X-t(1) - xi(t)|) 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. (C) 2017 Elsevier B.V. All rights reserved.