1. PRELUDE 3
Figure 1. λ = 0; ut(x) =
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 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)
for all t 0
and x ∈ [0 , 1]; that is, the solution at time t is equal to the initial value u0 times a
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.
1977, Philip Warren Anderson was awarded the Nobel Prize in Physics for this and
subsequent related works on disordered [i.e., random] systems.