Averaging principle for a class of stochastic reaction–diffusion equations

By Sandra Cerrai, Mark Freidlin. 2008

210 citations recorded by the source corpus.

We consider the averaging principle for stochastic reaction–diffusion equations. Under some assumptions providing existence of a unique invariant measure of the fast motion with the frozen slow component, we calculate limiting slow motion. The study of solvability of Kolmogorov equations in Hilbert spaces and the analysis of regularity properties of solutions, allow to generalize the classical approach to finite-dimensional problems of this type in the case of SPDE’s.

Subfields: Finance, Mathematical Physics, Modeling and Simulation

View Paper: https://www.semanticscholar.org/paper/06ab860c4f18bdbd728e4be1cc43b3756c399f40

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Citations in the corpus

Strong and weak orders in averaging for SPDEs (2012, methodology): “When working with SDEs, this strategy can be entirely followed; nevertheless in the case of SPDEs, the Kolmogorov equations involve the unbounded operators A and B, and quantities like Tr(D(2)yyu), and this leads to technical problems—see [3] or [5].”
Orders of convergence in the averaging principle for SPDEs: The case of a stochastically forced slow component (2018, methodology): “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.”
Elliptic and parabolic equations for measures (2009, background): “, the book [67]), especially connected with stochastic partial differential equations (see [11], [52], [57], [58], [62], [63], [70], [74], [79], [82], [90], [91], [128], [150], [163]), infinite gradient systems, Gibbs measures, and stochastic quantization (see [5], [6], [56], [83], [85], [86], [96],”
Two-time-scales hyperbolic–parabolic equations driven by Poisson random measures: Existence, uniqueness and averaging principles (2017, methodology): “Moreover, we will work in the framework of Green functions, which is a little different from the previous works [6, 7, 16, 18] investigating the coupled hyperbolic-parabolic equations.”
Strong convergence in stochastic averaging principle for two time-scales stochastic partial differential equations☆ (2011, methodology): “It was followed by [5] and [6] that the first author extended the results in the [4] to the general case, i.”