Pathwise estimates for an effective dynamics

2016/05/09 by Frederic Legoll, Frédéric Legoll, Legoll, Frederic +5 · 3 citations
Mathematics · #FOS: Mathematics #Probability (math.PR) #math.PR

paper · pdf · doi:10.48550/arxiv.1605.02644

arxiv created 2016/05/09 · arxiv updated 2016/05/10

Abstract

Starting from the overdamped Langevin dynamics in ℝⁿ, dX_t = -∇ V(X_t) dt + √2 β^-1 dW_t, we consider a scalar Markov process ξ_t which approximates the dynamics of the first component X¹_t. In the previous work [F. Legoll, T. Lelievre, Nonlinearity 2010], the fact that (ξ_t)_{t ≥ 0} is a good approximation of (X¹_t)_{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} | X¹_t - 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.

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• Generative modeling of conditional probability distributions on the level-sets of collective variables
• Consistent Projection of Langevin Dynamics: Preserving Thermodynamics and Kinetics in Coarse-Grained Models
• Learning collective variables that preserve transition rates

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