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:
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