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Pathwise estimates for an effective dynamics

Legoll, Frédéric; Lelièvre, Tony; Olla, Stefano (2017), Pathwise estimates for an effective dynamics, Stochastic Processes and their Applications, 127, 9, p. 2841-2863. 10.1016/j.spa.2017.01.001

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Type
Article accepté pour publication ou publié
Date
2017
Journal name
Stochastic Processes and their Applications
Volume
127
Number
9
Publisher
Elsevier
Pages
2841-2863
Publication identifier
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Author(s)
Legoll, Frédéric

Lelièvre, Tony cc

Olla, Stefano cc
CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
Abstract (EN)
Starting from the overdamped Langevin dynamics in Rn,dXt=−∇V(Xt)dt+2β−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(sup0≤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.
Subjects / Keywords
Effective dynamics for SDEs; Pathwise estimates; Averaging