Strong convergence of principle of averaging for multiscale stochastic dynamical systems

Liu, Di

doi:10.4310/cms.2010.v8.n4.a11

Cited by 90 publications

(62 citation statements)

References 7 publications

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“…For results concerning the averaging principle for Stochastic Differential Equations (SDEs), we refer to the pioneering work [24], and to the following extensions since then, e.g. [25], [26], [29], [35] (this list is not exhaustive).…”

Section: Introductionmentioning

confidence: 99%

Orders of convergence in the averaging principle for SPDEs: The case of a stochastically forced slow component

Bréhier

2020

Stochastic Processes and their Applications

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This article is devoted to the analysis of semilinear, parabolic, Stochastic Partial Differential Equations, with slow and fast time scales. Asymptotically, an averaging principle holds: the slow component converges to the solution of another semilinear, parabolic, SPDE, where the nonlinearity is averaged with respect to the invariant distribution of the fast process.We exhibit orders of convergence, in both strong and weak senses, in two relevant situations, depending on the spatial regularity of the fast process and on the covariance of the Wiener noise in the slow equation. In a very regular case, strong and weak orders are equal to 1 2 and 1. In a less regular case, the weak order is also twice the strong order.This study extends previous results concerning weak rates of convergence, where either no stochastic forcing term was included in the slow equation, or the covariance of the noise was extremely regular.An efficient numerical scheme, based on Heterogeneous Multiscale Methods, is briefly discussed.where the domain is D = (0, 1) d , for some d ∈ {1, 2, 3}, andẆ is a Gaussian noise which is white in time, and white or correlated in space. For a mathematically precise formulation, see (1), in the framework of [11], and Section 2. The fast component is given by y ǫ (t, ·) = y(ǫ −1 t, ·), and is Key words and phrases. Stochastic Partial Differential Equations, Averaging Principle, Poisson equation in infinite dimension, Heterogeneous Multiscale Method, strong and weak error estimates. 1 assumed to be an ergodic process, independent of the Wiener process W which appears in the slow equation, i.e. the equation for the slow component x ǫ .The averaging principle for such SPDE systems has been proved in a very general framework in [8], [10] for globally Lipschitz coefficients, and later in [9] in the case of non-globally Lipschitz continuous coefficients. In these results, no order of convergence, in terms of ǫ → 0, is provided. The first study of orders of convergence for the averaging principle in the SPDE case, was performed by the author in [3]. The main motivation for studying the rates of convergence is the construction of efficient numerical schemes, based on the Heterogeneous Multiscale Methods, see [4] and references therein.In recent years, many works have been devoted to the study of the averaging principle for different classes of SPDEs. For instance, see [12,15,16,19,20,28], in the parabolic SPDE case. See [17,18], for some parabolic-hyperbolic systems. See [21,22] in the Schrödinger equations case. Finally, see [1,2], where stochastic fluid mechanics equations are considered, with motivations coming from physics.As is usual when dealing with stochastic equations, orders of convergence are understood in two senses. First, strong convergence deals with the mean-square error. Second, weak convergence is related to convergence in distribution, considering sufficiently smooth test functions. If the averaging principle for SDEs (with globally Lipschitz continuous coefficients) is considered, the strong or...

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Section: Introductionmentioning

confidence: 99%

Orders of convergence in the averaging principle for SPDEs: The case of a stochastically forced slow component

Bréhier

2020

Stochastic Processes and their Applications

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“…With a slight abuse of notation, in (21) we have denoted by ∇ξ i the usual gradient of the function ξ i , for 1 ≤ i ≤ m. Assuming the vectors ∇ξ 1 , ∇ξ 2 , · · · , ∇ξ m are linearly independent (see Assumption 1 in Subsection 2.2), we have that Φ is positive definite and therefore invertible. In this case, we denote by A the positive definite symmetric matrix given by…”

Section: Notationsmentioning

confidence: 99%

Pathwise Estimates for Effective Dynamics: The Case of Nonlinear Vectorial Reaction Coordinates

Lelièvre

Zhang

2019

Multiscale Model. Simul.

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Effective dynamics using conditional expectation was proposed in [18] to approximate the essential dynamics of high-dimensional diffusion processes along a given reaction coordinate. The approximation error of the effective dynamics when it is used to approximate the behavior of the original dynamics has been considered in recent years. As a continuation of the previous work [19], in this paper we obtain pathwise estimates for effective dynamics when the reaction coordinate function is either nonlinear or vector-valued.

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“…Thanks to (2.7), by adapting the arguments used in [4] and [6], it is possible to show that there exists a unique invariant measure µ x for the semigroup P x t , which satisfies…”

Section: Ergodicity For the Fast Motion Equationmentioning

confidence: 99%

“…The main contribution of the present paper is to prove the L p (p > 2)-stronger convergence than the convergence in probability and mean-square shown in [11,12,7,3,13,5,4,6], which is crucial in numerical analysis and engineering fields. In addition, by Hölder's inequality, Chebyshev's inequality and our results, it is very easy to derive that the slow component can be approximated by solutions of the corresponding averaged equation in the sense of both convergence in L q (0 < q ≤ p)…”

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confidence: 98%

Lp(p>2)-strong convergence of an averaging principle for two-time-scales jump-diffusion stochastic differential equations

Miao

2015

Nonlinear Analysis: Hybrid Systems

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Strong convergence of principle of averaging for multiscale stochastic dynamical systems

Cited by 90 publications

References 7 publications

Orders of convergence in the averaging principle for SPDEs: The case of a stochastically forced slow component

Orders of convergence in the averaging principle for SPDEs: The case of a stochastically forced slow component

Pathwise Estimates for Effective Dynamics: The Case of Nonlinear Vectorial Reaction Coordinates

Lp(p>2)-strong convergence of an averaging principle for two-time-scales jump-diffusion stochastic differential equations

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