1. PRELUDE 3

Figure 1. λ = 0; ut(x) =

sin(πx)e−π2t/2

(1.2)

⎡

⎢

⎢

⎣

∂

∂t

ut(x) =

1

2

∂2

∂x2

ut(x) + λut(x)ξt(x) for all t 0 and x ∈ [0 , 1];

u0(x) = sin(πx) for all t 0 and x ∈ [0 , 1];

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

where λ ∈ R is a parameter that gauges the influence of the noise ξ on the solution u,

and we recall that ξ denotes space-time white noise. The stochastic PDE (1.2) is an

example of a parabolic Anderson model—so named after the 1958 milestone paper

of P.W. Anderson on condensed-matter physics [2]3—and serves as motivation for

some of the ensuing theory.

Because −ξ is a space-time white noise as well, we may—and will—only con-

sider λ 0 from now on. The solution with forcing constant λ 0 has the same

probability law as the one with −λ 0, after all.

Let us consider first the case that λ = 0. In that case, (1.2) reduces to the

non-random heat equation (1.1). And because we have selected the initial value

u0(x) = sin(πx) to be an eigenfunction of the Dirichlet Laplacian, we may compute

the solution explicitly: The solution is ut(x) = sin(πx)

exp(−π2t/2)

for all t 0

and x ∈ [0 , 1]; that is, the solution at time t is equal to the initial value u0 times a

dissipation term

exp(−π2t/2).

Figure 1 depicts a space-time numerical computation

of the solution to (1.2)—with λ = 0—run from time t = 0 to time t = 0.1.

3In

1977, Philip Warren Anderson was awarded the Nobel Prize in Physics for this and

subsequent related works on disordered [i.e., random] systems.