Pacific Institute for the Mathematical Sciences - PIMS

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 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)

3:30pm, WMAX 216