ABSTRACT
Consider the following random linear parabolic equation:
∂u
∂t =
∂
∂xi
( N∑ j=1
aij(θtω, x) ∂u
∂xj + ai(θtω, x)u
) +
bi(θtω, x) ∂u
∂xi + c0(θtω, x)u, x ∈ D,
Bω(t)u = 0, x ∈ ∂D,
(5.0.1)
where Bω(t) = Baω (t), Baω is the boundary operator in (2.0.3) with a being replaced by aω(t, x) = (aij(θtω, x), ai(θtω, x), bi(θtω, x), c0(θt, x), d0(θtω, x)), d0(ω, x) ≥ 0 for all ω ∈ Ω and a.e. x ∈ ∂D, ((Ω,F,P), {θt}t∈R) is an ergodic metric dynamical system, and the functions aij (i, j = 1, . . . , N), ai (i = 1, . . . , N), bi (i = 1, . . . , N) and c0 are (F × B(D),B(R))-measurable, and the function d0 is (F×B(∂D),B(R))-measurable; and consider the following nonautonomous linear parabolic equation:
∂u
∂t =
∂
∂xi
( N∑ j=1
aij(t, x) ∂u
∂xj + ai(t, x)u
) +
bi(t, x) ∂u
∂xi + c0(t, x)u, x ∈ D,
Ba(t)u = 0, x ∈ ∂D,
(5.0.2)
where Ba is the boundary operator in (2.0.3) with a(t, x) = (aij(t, x), ai(t, x), bi(t, x), c0(t, x), d0(t, x)), d0(t, x) ≥ 0 for a.e. (t, x) ∈ R× ∂D.
