abstract:
First lecture:
The first lecture will consist of an introduction to stochastic
integration with respect to Brownian motion, in particular on Hilbert
spaces. We shall try to keep prerequisites to a minimum as far as this is
possible.
Second lecture:
The second lecture will present the main existence and uniqueness result
for stochastic differential equations (SDE) under monotonicity conditions
within the so-called "variational approach". We shall also briefly recall
the special finite dimensional case, where the proof is based on the Euler
approximation. If time permits, we shall also compare this with other
approaches ("semigroup approach", "martingale approach").
Third lecture:
The third lecture will be devoted to examples and applications to
stochastic partial differential equations of evolutionary type. These
include the stochastic heat equation, the stochastic p-Laplace equation
and the stochastic porous media equation. In particular, the latter has
been analyzed recently in a number of papers in regard to various aspects.
If there is time left, I shall give a short overview about the respective
results.
The lectures will be based on the following reference:
C.Prevot, M. Roeckner: A Concise Course on Stochastic Partial Differential
Equations, Lecture Notes in Mathematics, 1905, Springer, Berlin 2007. |