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

Abstract

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...

The averaging principle for stochastic differential equations driven by a Wiener process revisited

“…That identity is due to the following observation: one has σ 2 `xσy 2 " σ 2 . The decomposition into average and fluctuation terms then clearly explains the diffusion enhancement in the averaged equation (2).…”

“…The averaging principle states that one can eliminate the fast process when ǫ Ñ 0, precisely the slow component X ǫ converges (in distribution) to the solution X of an autonomous evolution equation called the averaged equation. In the case of the system (1), the averaged equation is a SDE of the type (2) dXptq " ΣpXptqqdβptq, where Σ 2 p¨q " σ 2 p¨q " ż σp¨, mq 2 dµpmq, and µ denotes the invariant probability distribution of the fast ergodic process `mptq ˘tě0 .…”

“…Let us also mention [19], and the recent works [17,18]. In the last decade, the averaging principle has been extensively studied for systems of stochastic partial differential equations, see for instance [4,5], contributions of the author [1,2] and references therein. Recently Hairer and Li [9] have extended the averaging principle for SDE systems of the type (1) where the standard Brownian motion β is replaced by a fractional Brownian motion β H with Hurst index H ą 1{2: in Section 4 below we explain how the point of view developped in the present article is related to that generalization.…”

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“…Apart from the above motivations, the averaging principle itself is also theoretically interesting, which has been studied a lot in the literatures. For instance, the averaging principle for a general class of stochastic reaction–diffusion systems with two time scales was investigated by Cerrai and Freidlin; 15 see also other studies 16‐20 and the references therein. Subsequently, many authors have endeavored to derive the averaging principle for different types stochastic partial differential equation (PDE) (SPDE).…”