INVARIANCE PRINCIPLES FOR PARABOLIC EQUATIONS WITH RANDOM COEFFICIENTS

摘要

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 (1) partial derivative(t)u(epsilon)(t, x) = Sigma(0 less than or equal tokless than or equal to 2p) A(k)(t/epsilon, x, omega) partial derivative(x)(k)u(epsilon)(t, x), u(epsilon)(0, x) = phi(x) 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 (2) partial derivative(v)v(epsilon)(tau, x) = Sigma(0 less than or equal tokless than or equal to 2p epsilon A?k(tau, x, omega) partial derivative(x)(k)v(epsilon)(tau, x), v(epsilon)(0, x) = phi(x) through time change. Next, we impose on the coefficients of (1) a pointwise (in x and I) weak law of large numbers and a weak invariance principle (3) {epsilon(h) integral(0)(u-1) A(k)(x, s) - A(k)(0)(x) ds}(kless than or equal to 2p double right arrow {Theta?k}?(kless than or equal to 2p) in C([0, T], H-1), H-1 being a separable Hilbert space of functions and h is an element of (0, ii denoting a constant. (h > 1/2 allows for long range time dependence.) Then, under these extraordinarily general conditions, we infer the weak invariance principle epsilon(h-1)(u(epsilon) - u) double right arrow (y) over cap. It is the non-random, epsilon-homogeneous solution of (4) partial derivative(t)u(t, x) = Sigma(0 less than or equal tokless than or equal to 2p) A(k)(0)(x) partial derivative(x)(k)u(t, x), u(0, x) = phi(x) and (y) over cap mildly satisfies the linear stochastic partial differential equation (5) partial derivative(t) (y) over cap(t, x) = Sigma(kless than or equal to 2p) A(k)(0)(x) partial derivative(x)(k) (y) over cap(t, x) dt + Sigma(kless than or equal to 2p Theta?k(dt, x) partial derivative(x)(k)u(t, x). (C) 1997 Academic Press. [References: 22]