Moments, Intermittency, and Growth Indices for Nonlinear Stochastic PDE's with Rough Initial Conditions

In this thesis, we study several stochastic partial differential equations (SPDE’s) in the spatial domain R, driven by multiplicative space-time white noise. We are interested in how rough and unbounded initial data affect the random field solution and the asymptotic properties of this solution. We first study the nonlinear stochastic heat equation. 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 non-compact support and even be non-tempered (for instance with exponentially growing tails). Existence and uniqueness is proved without appealing to Gronwall’s lemma, by keeping tight control over moments in the Picard iteration scheme. Upper and lower bounds on all p-th moments (p ≥ 2) are obtained. These bounds become equalities for the parabolic Anderson model when p = 2. We determine the growth indices introduced by Conus and Khoshnevisan [19] and, despite the irregular initial conditions, we establish Hölder continuity of the solution for t > 0. In order to study a wider class of SPDE’s, we consider a more general problem, con- sisting in a stochastic integral equation of space-time convolution type. We give a set of assumptions which guarantee that the stochastic integral equation in question has a unique random field solution, with moment formulas and sample path continuity properties. As a first application, we show how certain properties of an extra potential term in the nonlinear stochastic heat equation influence the admissible initial data. As a second application, we investigate the nonlinear stochastic wave equation on R+ × R. All the properties obtained for the stochastic heat equation – moment formulas, growth indices, Hölder continuity, etc. – are also obtained for the stochastic wave equation.