Invariance Principles for Parabolic Equations with Random Coefficients

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|⩽2p Ak(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|⩽2p εAk(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−10 Ak(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|⩽2p A0k(x) ∂kxu(t, x), u(0, x)=ϕ(x) ((4))andŷ; mildly satisfies the linear stochastic partial differential equation∂tŷ(t, x)=∑|k|⩽2p A0k(x) ∂kxŷ(t, x) dt+∑|k|⩽2p Θk(dt, x) ∂kxu(t, x). ((5))