Moment bounds in spde's with application to the stochastic wave equation

Abstract

We exhibit a class of properties of an spde that guarantees existence, uniqueness and bounds on moments of the solution. These moment bounds are expressed in terms of quantities related to the associated deterministic homogeneous p.d.e. With these, we can, for instance, obtain solutions to the stochastic heat equation on the real line for initial data that falls in a certain class of Schwartz distributions, but our main focus is the stochastic wave equation on the real line with irregular initial data. We give bounds on higher moments, and for the hyperbolic Anderson model, explicit formulas for second moments. We establish weak intermittency and obtain sharp bounds on exponential growth indices for certain classes of initial conditions with unbounded support. Finally, we relate H\"older-continuity properties of the stochastic integral part of the solution to the stochastic wave equation to integrability properties of the initial data, obtaining the optimal H\"older exponent.