Introduction to Stochastic Partial Diverential Equations

Timetable
Lunedì 13, 20 e 27 marzo: 14.30 -17.30 aula M

Course description:
An introduction to Stochastic Partial Di erential Equations (SPDEs) with applications.
Course structure
1. Gaussian measure theory
Random vectors and Bochner integral. Some elements of probability in in nite-dimensional
spaces are considered, with emphasis on the integration of random vectors with values in
separable Banach-spaces and in operator spaces.
Gaussian measures. We introduce cylindrical Gaussian random variables and Hilbert-spacevalued
Gaussian random variables and then de ne cylindrical Wiener processes and Q􀀀Wiener
processes (i.e. with the covariance given by the trace-class operator Q) in a natural way.
Stochastic integral and It^o's formula. The stochastic integral is constructed with respect
to a cylindrical Wiener process, then with respect to a Q􀀀Wiener process, by extending the
integral of elementary processes. Some properties of the stochastic integral are given, including
It^o's formula.
2. Stochastic Di erential Equations
Semigroup Theory. In this section we review the fundamentals of semigroup theory.
Stochastic Convolutions and Linear SPDEs. We derive existence and uniqueness of di erent
types of solutions for linear SDEs driven by generators of C0-semigroups. The method is based
on the study of the stochastic convolution.
Solutions by Variational Method. The purpose is to study solutions of nonlinear SPDEs,
which are seen as evolution equations in a Gelfand triplet, under assumptions of compact
embedding or monotone coecients.
3. Applications
Along the abstract study of SDEs in in nite-dimensional spaces, various examples of SPDEs
with applications in physics, biology or mathematical nance will be given.
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Selected Bibliography:
Da Prato, G., An introduction to in nite dimensional analysis. Appunti dei Corsi Tenuti da
Docenti della Scuola. Scuola Normale Superiore, Pisa, 2001.
Da Prato, G.; Zabczyk, J., Stochastic equations in in nite dimensions. Second edition. Encyclopedia
of Mathematics and its Applications, 152. Cambridge University Press, 2014.
Gawarecki, L.; Mandrekar, V., Stochastic di erential equations in in nite dimensions with ap-
plications to stochastic partial di erential equations. Probability and its Applications. Springer,
2011.
Liu, W.; Rockner, M., Stochastic partial di erential equations: an introduction. Universitext.
Springer, 2015.
Prev^ot, C.; Rockner, M., A concise course on stochastic partial di erential equations. Lecture
Notes in Mathematics, Springer, 2007.
Zabczyk, J., Topics in stochastic processes. Scuola Normale Superiore di Pisa. Quaderni.
Scuola Normale Superiore, Pisa, 2004.
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Il materiale ed i video delle lezioni del seminario sono raccolti nell'area e-Learning del corso Stochastic differential equations (2016/2017)
https://moodle.univr.it/moodle/course/view.php?id=1026