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Pathwise estimates for an effective dynamics

Abstract : Starting from the overdamped Langevin dynamics in $\mathbb{R}^n$, $$ dX_t = -\nabla V(X_t) dt + \sqrt{2 \beta^{-1}} dW_t, $$ we consider a scalar Markov process $\xi_t$ which approximates the dynamics of the first component $X^1_t$. In the previous work [F. Legoll, T. Lelievre, Nonlinearity 2010], the fact that $(\xi_t)_{t \ge 0}$ is a good approximation of $(X^1_t)_{t \ge 0}$ 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 $\mathbb{E} \left( \sup_{0 \leq t \leq T} \left| X^1_t - \xi_t \right| \right)$, 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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https://hal.archives-ouvertes.fr/hal-01314221
Contributor : Frederic Legoll <>
Submitted on : Wednesday, May 11, 2016 - 12:22:03 AM
Last modification on : Monday, December 14, 2020 - 5:25:05 PM

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  • HAL Id : hal-01314221, version 1
  • ARXIV : 1605.02644

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Frédéric Legoll, Tony Lelièvre, Stefano Olla. Pathwise estimates for an effective dynamics. Stochastic Processes and their Applications, Elsevier, 2017, 127 (9), pp.2841-2863. ⟨hal-01314221⟩

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