4 1. PRELUDE

Figure 2. λ = 0.1

One might imagine that when λ 0, the noise ξ has a certain amount of

influence on the behavior and properties of the solution u. And that the influence

of the noise ought to increase as λ is increased. Figure 2 depicts a simulation in

the case that λ = 0.1.

For purposes of comparison, all of our computations and/or simulations are run

from t = 0 to t =

0.1.4

Now λ = 0.1 amounts to a very small amount of noise, and we can see that

the solution to (1.2) is essentially a small random perturbation of the non-random

case that λ = 0; see Figure 2. Upon closer examination, however, we find the

somewhat surprising fact that the maximum peak of the simulated solution—which

is manifestly 1 when λ = 0—is now unduly large [≈ 1.4, though it is a little hard

to see that in this figure] when λ = 0.1.

As it turns out, further increases in λ induce tremendous changes in the qual-

itative behavior of the solution. Figure 3 shows a simulation of (1.2) when λ = 2;

this value of λ represents a modest amount of noisy forcing. And when λ is rel-

atively large, say λ = 5, then the solution to (1.2) looks quite different from the

non-random case λ = 0; see Figure 4 for a typical simulation.

In the present setting of SPDEs, “intermittency” is a term for the property

that, given enough noise, the solution develops very tall peaks. In Figures 3 and 4

the simulated maximum peaks are respectively at about 35 and 2.5×1019! In recent

collaboration with Kunwoo Kim [72, 73], we have developed a theory that explains

4These

and the subsequent simulations are borrowed from the recent work of Khoshnevisan

and Kim [72], where you can find mathematical explanations of the phenomena that we are about

to discover by experimentation.