2 1. PRELUDE

to keep the endpoints permanently at zero flux, then ut(0) = ut(1) = 0 [Dirichlet

boundary conditions], and (1.1) has a unique classical solution.

In the presence of an external cooling/heating source, the heat equation (1.1)

has to be revised as follows:

⎡

⎣

∂

∂t

ut(x) = ν

∂2

∂x2

ut(x) + ft(x) for all t 0 and x ∈ [0 , 1],

ut(0) = ut(1) = 0 for all t 0,

where ft(x) denotes the density of the forcing. The forcing itself may be a func-

tion of the solution u. In the simplest non-trivial case, one obtains the following

boundary-value problem with “multiplicative forcing”:

⎡

⎣

∂

∂t

ut(x) = ν

∂2

∂x2

ut(x) + σ(ut(x))Ft(x) for all t 0 and x ∈ [0 , 1],

ut(0) = ut(1) = 0 for all t 0,

where Ft(x) denotes now [the exogenous] forcing density, and σ : R → R describes

the nature of the feedback mechanism between the solution u and the forcing term

F .

The stochastic PDEs that are studied in these lectures are examples of partial

differential equations of the preceding general type, with the additional restriction

that Ft(x) is a two-parameter stochastic process—hereforth, a random field—or

what is more commonly the case, F is a “random noise,” also known as a “gener-

alized random field.” When the solution to the resulting stochastic PDE exists, it

models heat density in a random medium.

Such models capture the essence of a great number of complex phenomena in

science and engineering; we will say a little about some of this vast literature later

on. In the mean time, I mention in passing that there are also many other stochastic

PDEs—that we do not treat here at all—that arise prominently in other contexts.

For instance, we might consider instead: (i) Stochastic wave equations, wherein

∂u/∂t can be replaced by

∂2u/∂t2;

(ii) Random scatterers, where the constant ν

can be replaced by ψ(ut(x)), where ψ is a random field; and (iii) Random automata,

where the initial function u0 is random.

Our analysis of SPDEs is specialized further to the case that F denotes “space-

time white noise.” Roughly speaking, this means that F is a centered Gaussian

process with covariance

Cov(Ft(x) , Fs(y)) = δ0(t − s)δ0(y − x).

Throughout these notes we will write ξ instead of F := space-time white noise; that

is, ξ is reserved especially for space-time white noise and nothing else.

Because “delta functions” are not bona fide functions, our casual covariance

description of ξ is not quite correct; a complete introduction requires more care, and

we will tend to this matter in the next chapter. Nevertheless, one may think about

ξ intuitively as follow: If (t , x) = (s , y), then ξt(x) and ξs(y) are “independent

Gaussian quantities.” The corresponding SPDEs arise in applications of probability

as PDEs that are driven by “zero-range random noises” or idealized approximations

to “very short-range-dependent physical noises.”

We now return to the following somewhat simplified stochastic boundary-value

problem: