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.
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