ABSTRACT
In this chapter, we shall study stochastic partial differential equations of parabolic type, or simply, stochastic parabolic equations in a bounded domain. As a relatively simple problem, consider the heat conduction over a thin rod (0 ≤ x ≤ L) with a constant thermal diffusivity κ > 0. Let u(x, t) denote the temperature distribution at point x and time t > 0 due to a randomly fluctuating heat source q(x, t, ω). Suppose both ends of the rod are maintained at the freezing temperature. Then, given an initial temperature u0(x), the temperature field u(x, t) is governed by the initial-boundary value problem for the stochastic heat equation:
∂u(x, t)
∂t = κ
∂2u
∂x2 + q(x, t, ω), x ∈ (0, L), t > 0,
u(0, t) = u(L, t) = 0,
u(x, 0) = u0(x).
