Preprint: ON THE STOCHASTIC HEAT EQUATION WITH SPATIALLY-COLORED RANDOM FORCING

Mohammud Foondun and Davar Khoshnevisan

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 u0 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