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Analysis and Numerics of Stochastic PDEs

Analysis and Numerics of Stochastic PDEs by Istvan Gyongy

This course is run by MIGSAA. It is open to everyone, including non-MIGSAA students, as an SMSTC supplementary module.

Semester 1, 2 hours over 10 weeks, 15 credits; assessment: solving assignment problems

In the first part of the lectures basic methods of solving stochastic partial differential
equations (SPDEs) of parabolic type will be presented. In particular, main results of
the L_2 and L_p theories for SPDEs given in the whole Euclidean space will be summarised,
and the theory of SPDEs given on domains will be presented in more details.
In the second part of the lectures methods for solving SPDEs numerically will be studied,
theorems on accuracy of numerical schemes will be proved.

Applications from population genetics and stochastic filtering will be discussed.

I. Main results on solvability of linear SPDEs
1. Stochastic processes with values in Sobolev spaces, and Ito formulas for their functions
2. Existence and uniqueness theorems in Sobolev spaces for SPDEs in the whole Euclidean
3. Solvability in weighted Sobolev spaces of SPDEs on bounded domains
4. Stochastic Fubini theorem and Ito-Wentzell formula. Feynman-Kac formulas for PDEs
and SPDEs
5. SPDEs in population genetics

II. Numerical schemes for PDEs and SPDEs of parabolic type
1. Spatial discretisation, rate of convergence, accelerated schemes
2. Accuracy of implicit and explicit time discretisation
3. Fully discretised schemes
4. Localisation error
5. Splitting up approximations, accelerated splitting up schemes
6. Wong-Zakai approximations for SPDEs

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