Invariance Principles for Parabolic Equations with Random Coefficients

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  • A general Hilbert-space-based stochastic averaging theory is brought forth herein for arbitrary-order parabolic equations with (possibly long range dependent) random coefficients. We use regularity conditions onView the MathML sourcewhich are slightly stronger than those required to prove pathwise existence and uniqueness for (1). Equation (1) can be obtained from the singularly perturbed systemView the MathML sourcethrough time change. Next, we impose on the coefficients of (1) a pointwise (inxandt) weak law of large numbers and a weak invariance principleView the MathML sourceinC([0, T], View the MathML source1), View the MathML source1being a separable Hilbert space of functions andh∈(0, 1) denoting a constant. (h>1/2 allows for long range time dependence.) Then, under these extraordinarily general conditions, we infer the weak invariance principleεh−1(uε−u)⇒ŷ.uis the non-random,ε-homogeneous solution ofView the MathML sourceandŷ; mildly satisfies the linear stochastic partial differential equationView the MathML source

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    Article (Published)
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    © 1997 Journal of Functional Analysis. This version of this article is open access and can be downloaded and shared. The original author(s) and source must be cited. Non-commercial use only.
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    • Donald A. Dawson, Michael A. Kouritzin, Invariance Principles for Parabolic Equations with Random Coefficients, Journal of Functional Analysis, Volume 149, Issue 2, 1 October 1997, Pages 377-414, ISSN 0022-1236,