2009 Probability Seminar - 11

The main topic of this talk is the non-linear stochastic wave equation in spatial dimension greater than 3 driven by spatially homogeneous Gaussian noise that is white in time. In dimensions greater than 3, the fundamental solution of the wave equation is neither a function nor a non-negative measure, but a general Schwartz distribution. Hence, we rst develop an extension of the Dalang-Walsh stochastic integral that makes it possible to integrate a wide class of Schwartz distributions. This class contains the fundamental solution of the wave equation. With this extended stochastic integral, we establish existence of a square-integrable random-eld solution to the non-linear stochastic wave equation in any dimension. Uniqueness of the solution is established within a specic class of processes. In the case of ane multiplicative noise, we obtain a series representation of the solution and estimates on the p-th moments of the solution (p > 1). From this, we deduce Holder-continuity of the solution. The Holder exponent that we obtain is optimal. For the case of general multiplicative noise, we construct a framework for working with appropriate iterated stochastic integrals and then derive a truncated It^o-Taylor expansion for the solution of the stochastic wave equation. The convergence of this expansion remains an open problem. (Joint work with Robert C. Dalang, Swiss Federal Institute of Technology)