Pathwise estimates for an effective dynamics

Frédéric Legoll; Tony Lelièvre; Stefano Olla

doi:10.1016/j.spa.2017.01.001

Pathwise estimates for an effective dynamics

Frédéric Legoll
, Tony Lelièvre
, Stefano Olla

Université Paris Est, ENPC LIGM, IMAGINE
Université Paris Dauphine

Research output: Contribution to journal › Article › peer-review

Abstract

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.

Original language	English
Pages (from-to)	2841-2863
Number of pages	23
Journal	Stochastic Processes and their Applications
Volume	127
Issue number	9
DOIs	https://doi.org/10.1016/j.spa.2017.01.001
Publication status	Published - 1 Sept 2017

Keywords

Averaging
Effective dynamics for SDEs
Pathwise estimates

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10.1016/j.spa.2017.01.001

Cite this

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title = "Pathwise estimates for an effective dynamics",

abstract = "Starting from the overdamped Langevin dynamics in Rn, dXt=−∇V(Xt)dt+2β−1dWt we consider a scalar Markov process ξt which approximates the dynamics of the first component Xt1. In the previous work (Legoll and Leli{\`e}vre, 2010), the fact that (ξt)t≥0 is a good approximation of (Xt1)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|Xt1−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.",

keywords = "Averaging, Effective dynamics for SDEs, Pathwise estimates",

author = "Fr{\'e}d{\'e}ric Legoll and Tony Leli{\`e}vre and Stefano Olla",

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year = "2017",

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doi = "10.1016/j.spa.2017.01.001",

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TY - JOUR

T1 - Pathwise estimates for an effective dynamics

AU - Legoll, Frédéric

AU - Lelièvre, Tony

AU - Olla, Stefano

PY - 2017/9/1

Y1 - 2017/9/1

N2 - Starting from the overdamped Langevin dynamics in Rn, dXt=−∇V(Xt)dt+2β−1dWt we consider a scalar Markov process ξt which approximates the dynamics of the first component Xt1. In the previous work (Legoll and Lelièvre, 2010), the fact that (ξt)t≥0 is a good approximation of (Xt1)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|Xt1−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.

AB - Starting from the overdamped Langevin dynamics in Rn, dXt=−∇V(Xt)dt+2β−1dWt we consider a scalar Markov process ξt which approximates the dynamics of the first component Xt1. In the previous work (Legoll and Lelièvre, 2010), the fact that (ξt)t≥0 is a good approximation of (Xt1)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|Xt1−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.

KW - Averaging

KW - Effective dynamics for SDEs

KW - Pathwise estimates

UR - https://www.scopus.com/pages/publications/85011311764

U2 - 10.1016/j.spa.2017.01.001

DO - 10.1016/j.spa.2017.01.001

M3 - Article

AN - SCOPUS:85011311764

SN - 0304-4149

VL - 127

SP - 2841

EP - 2863

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 9

ER -

Pathwise estimates for an effective dynamics

Abstract

Keywords

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