ABSTRACT
In finite dimensions, it is well known that the solution of an Itoˆ equation is a diffusion process and the expectation of a smooth function of the solution process satisfies a diffusion equation in Rd, known as the Kolmogorov equation. Therefore it is quite natural to explore such a relationship for the stochastic partial differential equations. For example, consider the randomly perturbed heat equation (3.34). By means of the eigenfunction expansion, the solution is given by u(·, t) = ∑∞k=1 ukt ek, where ukt , k = 1, 2, · · · , satisfy the infinite system of Itoˆ equations (3.39) in [0, T ]:
dukt = −λkukt dt+ σkdwkt , uk0 = hk, k = 1, 2, ... Let un(·, t) = ∑nk=1 ukt ek be the n-term approximation of u. Then un(·, t) is an Ornstein-Uhlenbeck process, a time-homogeneous diffusion process in Rn. Let (y1, · · · , yn) ∈ Rn. Then, for a smooth function Φ on Rn, the expectation function Ψ(y1, · · · , yn, t) = E {Φ(u1t , · · · , unt )|u10 = y1, · · · , un0 = yn} satisfies the Kolmogorov equation:
∂Ψ
∂t = AnΨ, Ψ(y1, · · · , yn, 0) = Φ(y1, · · · , yn), (9.1)
where An is the infinitesimal generator of the diffusion process un(·, t) given by
An = 1 2
∂y2k −
λkyk ∂
∂yk .
