Lecture Series : Singular stochastic PDEs
Willem van Zuijlen (Weierstrass Institute, Berlin)
| Speaker |
Willem van Zuijlen (Weierstrass Institute, Berlin)
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|---|---|
| When |
Mar 20, 2023 03:30 PM
to
Mar 31, 2023 04:30 PM |
| Where | via zoom |
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Abstract: In this mini course we consider the paracontrolled distribution approach to solve singular stochastic partial differential equations (SPDEs) that is developed by Gubinelli, Imkeller and Perkowski. This theory, like the theory of regularity structures, is motivated by the Rough Path method that Terry Lyons invented in the 90s in order to solve stochastic differential equations in a pathwise sense. SPDEs are partial differential equations with random coefficients. They are called singular if one cannot make sense of the equation in a classical sense. In various examples of SPDEs, the white noise is part of the equation, which is irregular, namely, it is not a function but a distribution of negative regularity. Therefore this may cause terms in the equation to be -a priori- ill-defined, often due to the appearance of a product of distributions, which is in general not defined.
The theory relies on harmonic analysis, in particular its notion of Besov spaces. We start by introducing these spaces and describe how we can multiply certain distributions in such spaces by means of the para- and resonance product. Then we demonstrate the strategy behind the paracontrolled distribution approach by considering the PHI4d equation $\Phi^4_d$, where $d$ is the dimension that we consider, which will be $1,2$ and $3$, in the same order of difficulty.
Lectures series on March 20, 22, 24, 27, 29, 31 from 3:30pm onward via zoom