1. PRELUDE 5

Figure 3. λ = 2

many aspects of these “nonlinear noise excitation” phenomena. In particular, it is

shown in [73] that simulations such as these generally tend to greatly underestimate

the actual peak sizes.

We can summarize these remarks as follows: The introduction of noise in some

partial differential equations can bring about not a small perturbation, but truly-

fundamental changes to the system that the underlying PDE is attempting to de-

scribe. We will see many more examples of the phenomenon “SPDE⊆PDE” scat-

tered throughout these lectures. If we are going to learn only one lesson from these

examples, then perhaps it should be that the theory of stochastic partial differential

equations, though currently still at its early stages of mathematical development,

already points to wholly-new exciting scientific opportunities and directions of re-

search for PDEs that describe dynamics in random media.

The structure of these lecture notes are as follows: In Chapter 2 we introduce

the general theory of stochastic integrals ` a la Norbert Wiener; see, for example [141]

and especially [142]. We use Wiener’s theory in order to introduce linear SPDEs

in Chapter 3. In Chapter 4 we follow the theory of Walsh [138] and Dalang [34] in

order to introduce the sort of infinite-dimensional Itˆ o-type stochastic integral that

we will use to study nonlinear SPDEs. There are several other ways of studying

SPDEs; see, for example the theories of Da Prato and Zabczyk [32, 31, 30], Krylov

and Rozovskii [83, 80, 81, 79], and Pr´ evˆ ot and R¨ ockner [115]. While each of the

different approaches specializes at successfully addressing different questions, all of

these approaches ultimately agree for almost all models of general interest; see the

recent work of Dalang and Quer–Sardanyons [36] for a careful statement of this