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-
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
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
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
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
ut(x) = ν
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
If, in addition, we dampen the endpoints of the rod in order
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.
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.