On the stochastic heat equation with spatially-colored random forcing

By Mohammud Foondun, Davar Khoshnevisan. 2010

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We consider the stochastic heat equation of the following form ∂ ∂t ut(x) = (Lut)(x) + b(ut(x)) + σ(ut(x))Ḟt(x) for t > 0, x ∈ R, where L is the generator of a Levy process and Ḟ 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 Lu is replaced by its massive/dispersive analogue Lu − λu where λ ∈ R. And we describe accurately the effect of the parameter λ on the intermittence of the solution in the case that σ(u) is proportional to u [the “parabolic Anderson model”]. We also look at the linearized version of our stochastic PDE, that is the case when σ 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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0de1fcee890d6a5069ea9e1f1d4e2e355b295866