2009 Probability Seminar - 11

  • Date: 04/23/2009
Lecturer(s):
Daniel Conus (University of Utah)
Location: 

University of British Columbia

Topic: 

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

Description: 

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)

Schedule: 

3:30pm, WMAX 216

Sponsor: 

pims