Invariance Principles for Parabolic Equations with Random Coefficients

Abstract

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 on∂tuε(t,x)=∑0⩽|k|⩽2pAk(t/ε,x,ω)∂kxuε(t,x),uε(0,x)=ϕ(x) ((1))which are slightly stronger than those required to prove pathwise existence and uniqueness for (1). Equation (1) can be obtained from the singularly perturbed system∂τvε(τ,x)=∑0⩽|k|⩽2pεAk(τ,x,ω)∂kxvε(τ,x),vε(0,x)=ϕ(x) ((2))through time change. Next, we impose on the coefficients of (1) a pointwise (inxandt) weak law of large numbers and a weak invariance principleεh∫tε−10Ak(x,s)−A0k(x)ds|k|⩽2p⇒{Θk}|k|⩽2p ((3))inC([0,T],H1), H1being 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 of∂tu(t,x)=∑0⩽|k|⩽2pA0k(x)∂kxu(t,x),u(0,x)=ϕ(x) ((4))andŷ; mildly satisfies the linear stochastic partial differential equation∂tŷ(t,x)=∑|k|⩽2pA0k(x)∂kxŷ(t,x)dt+∑|k|⩽2pΘk(dt,x)∂kxu(t,x). ((5))