## 2009 Probability Seminar - 11

- Date: 04/23/2009

University of British Columbia

The non-linear wave equation in high dimensions : existence, Hölder-continuity and Itô-Taylor expansion.

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 specic 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)

3:30pm, WMAX 216