Moments and Growth Indices for the Nonlinear Stochastic Heat Equation with Rough Initial Conditions1 by Le Chen

We study the nonlinear stochastic heat equation in the spatial domain R, driven by space–time white noise. A central special case is the parabolic Anderson model. The initial condition is taken to be a measure on R, such as the Dirac delta function, but this measure may also have noncompact support and even be nontempered (e.g., with exponentially growing tails). Existence and uniqueness of a random field solution is proved without appealing to Gronwall’s lemma, by keeping tight control over moments in the Picard iteration scheme. Upper bounds on all pth moments (p ≥ 2) are obtained as well as a lower bound on second moments. These bounds become equalities for the parabolic Anderson model when p = 2. We determine the growth indices introduced by Conus and Khoshnevisan [Probab. Theory Related Fields 152 (2012) 681–701].