CHAPTER 1

Prelude

This is a 10-lecture course on the quantitative analysis of various parabolic

stochastic partial differential equations—written as parabolic SPDEs, to be short—

with special emphasis paid to a property that is sometimes referred to as “inter-

mittency.”

The general area of SPDEs is interesting to mathematicians because it contains

an enormous number of challenging open problems. There is also a great deal of

interest in this topic because it has deep applications in disciplines that range from

applied mathematics, statistical mechanics, and theoretical physics, to theoretical

neuroscience, theory of complex chemical reactions [including polymer science], fluid

dynamics, and mathematical finance. In fact, the combined literature on pure and

applied SPDEs is so big that it would require many more pages than I have in order

to include a proper bibliography. And to be perfectly honest, it also would require

more expertise than I have to discuss all of this enormous literature in suﬃcient

detail.

Instead of carrying out an exhaustive discussion, let us then begin with an

intuitive description of some of the kinds of problems that we plan to discuss in

these lectures.

Let us consider heat flow in a rod of unit length. We assume that the rod is

idealized: It is infinitesimally thin, as well as perfectly homogeneous. Thus, we may

identify it with the unit interval [0 , 1].

Suppose that, at time t = 0, the rod is at a certain temperature, and let us

denote the heat density, or thermal flux, at position x ∈ [0 , 1] by u0(x). This is

another way of saying that the total amount of heat in any subinterval [a , b] of the

rod [0 , 1] is

b

a

u0(x) dx at time t = 0.

According to the Fourier law of classical thermodynamics, if there are no exter-

nal heat/cooling sources, then heat flux evolves with time according to the linear

heat equation. That is, if ut(x) denotes the heat density, or thermal flux, at time

t at position x ∈ [0 , 1] along the rod, then the function u has two variables, and

solves the partial differential

equation,1

(1.1)

∂

∂t

ut(x) = ν

∂2

∂x2

ut(x) for all t 0 and x ∈ (0 , 1).

The constant ν 0 is the so-called diffusitivity of the rod, and describes the rod’s

thermal

properties.2

If, in addition, we dampen the endpoints of the rod in order

1Caveat:

We are using probabilists’ notation here: ut(x) denotes the evaluation of the

function u at the space-time point (t , x). This notation makes good sense in the present setting,

since we will be interested mainly in the evolution of the infinite-dimensional dynamical system

t → ut. We will never write ut in place of the time derivative of u.

2One

can write ν := C/(DH), where C, D, and H respectively denote the thermal conduc-

tivity, mass density, and the specific-heat capacity of the rod.

1

http://dx.doi.org/10.1090/cbms/119/01