Module 24-M-PT-SPDE Stochastic Partial Differential Equations

Non-official translation of the module descriptions. Only the German version is legally binding.

Students master advanced content and methods of the theory ofStochastic Partial Differential Equations, in particular they can independently carry out very complex proofs in this area requiring a high level of mathematical expertise.They are able to define central concepts of the theory (e.g. the stochastic integral in Hilbert spaces) and apply them in context. They can carry out existence and uniqueness proofs for stochastic differential equations and apply the theory to stochastic partial differential equations. They can use examples to illustrate concepts and theorems.
Students will be introduced to current research questions in the area of Stochastic Partial Differential Equations. They are able to recognise and assess further development opportunities and research goals.
Furthermore, students recognise further-reaching connections to mathematical issues that have already been worked out. They can transfer and apply the knowledge and methods they have learnt so far to deeper mathematical problem areas. Students also expand their mathematical intuition as a result of more intensive study.
In combination with other in-depth modules, they will be able to write their own research papers, e.g. a master's thesis in the field of Stochastic Partial Differential Equations.
In the tutorials, students develop their ability to discuss mathematical topics and thus further prepare themselves for the requirements of the Master's module, in particular for the scientific discussion within the Master's seminar presentation and the defence of their Master's thesis.

The following advanced content of teaching from the field of Stochastic Partial Differential Equations is compulsory:

1. Stochastic Integral in Hilbert Spaces: Infinite-dimensional Wiener processes, Martingales in general Banach spaces, Construction and Properties of the stochastic Integral, the stochastic integral for cylindrical Wiener processes
2. SDEs in Finite Dimensions: Existence and uniqueness of solutions, localization lemma
3. SDEs in Infinite Dimensions and Applications to SPDEs: Gelfand triples, conditions on the coefficients and examples, existence and Uniquness of solutions and an Itô formula, Markov property and invariant measures

This module prepares the content of a master's thesis.

Solid knowledge of stochastic analysis (24-M-PT-STA)

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