Pathwise estimates for an effective dynamics

الأصل · EN

Starting from the overdamped Langevin dynamics in Rⁿ, dXₜ = -∇ V(Xₜ) dt + √2 β⁻¹ dWₜ, we consider a scalar Markov process ξₜ which approximates the dynamics of the first component X¹ₜ. In the previous work [F. Legoll, T. Lelievre, Nonlinearity 2010], the fact that (ξₜ)ₜ ≥ ₀ is a good approximation of (X¹ₜ)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 (₀ ≤ ₜ ≤ ₜ | X¹ₜ - 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.