ON THE STOCHASTIC HEAT EQUATION WITH SPATIALLY-COLORED RANDOM FORCING

**Abstract.**
We consider the stochastic heat equation of the following form
$$
\partial_t u_t(x) = (\mathcal{L}u_t)(x) + \sigma(u_t(x))\partial_{tx}F_t(x)
\qquad\hbox{for $t>0$, $x\in{\bf R}^d$},
$$
where \(\mathcal{L}\)
is the generator of a Lévy process and \(\partial_{tx}F\)
is a spatially-colored,
temporally white, gaussian noise. We will be concerned mainly with the long-term
behavior of the mild solution to this stochastic PDE.
For the most part, we work under the assumptions that the initial data
u_{0} is a bounded and measurable function and σ is
nonconstant and Lipschitz continuous.
In this case, we find conditions under which the preceding stochastic PDE admits a unique
solution which is also *weakly intermittent*. In addition, we study the same
equation in the case that £u is replaced by its
massive/dispersive analogue \(\mathcal{L}u-\lambda u\)
where \(\lambda\in{\bf R}\).
And we describe accurately the effect of the parameter \(\lambda\) on
the intermittence of the solution in the case that \(\sigma(u)\) is proportional
to \(u\) [the ``parabolic Anderson model''].
Finally, we look at the linearized version of our stochastic PDE, that is
the case when \(\sigma\) is identically equal to one [any other constant
works also].
In this case, we study not only the existence and uniqueness of a solution, but
also the regularity of the solution when it exists and is unique.

**Keywords.**
Stochastic heat equation, intermittency.

**
AMS Classification (2000).**
Primary. 60H15; Secondary. 35R60.

**Support.** The research of DK was supported in part by grant DMS-0706728 from
the U.S. National Science Foundation.

**Pre/E-Prints.** This paper is available in

Mohammud Foondun Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. mohammud@math.utah.edu |
Davar Khoshnevisan Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu |

*Last Update: February 25, 2010*

©
2010 - Mohammud Foondun and Davar Khoshnevisan