Pathwise estimates for an effective dynamics

Starting from the overdamped Langevin dynamics in Rⁿ, dX_t=−∇V(X_t)dt+2β⁻¹dW_t we consider a scalar Markov process ξ_t which approximates the dynamics of the first component X_t¹. In the previous work (Legoll and Lelièvre, 2010), the fact that (ξ_t)_t≥0 is a good approximation of (X_t¹)_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_t¹−ξ_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.