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Talk by Anno Kurth

Rheinische Friedrich-Wilhelms-Universität Bonn

29 Jun 2018 10:30
29 Jun 2018
Building 15.22, E1, Room 3013 (INM-6 Library), First floor

Singular SPDEs, Paracontrolled Calculus and Mild Solutions

In my talk, I will deal with a technique to solve certain highly singular
stochastic partial di↵erential equations exemplified by the Parabolic Anderson
Model (PAM) formally written as the Cauchy problem.
I will briefly introduce the basic concepts of a solution theory for this kind
of equations called paracontrolled calculus. This theory builds on
Bony’s notion of paraproduct to handle the singular nature of certain SPDEs
by means of a renormalization. The introduced framework stands out to
other solution theories due to on the one hand being powerful enough to
treat a wide class of singular SPDEs and on the other hand being relatively
lightweight as well as connecting stochastics with the modern study of PDEs.
As it turns out, the paracontrolled calculus is based on the notion of weak
solution of PDEs. There are, however, also other concepts generalizing the
classical notion of solution to a PDE. In my master thesis I worked on a
paracontrolled calculus based on such a di↵erent concept called mild solution.
I will introduce the basics of this ’mild paracontrolled calculus’ and
show how to obtain a solution theory analogous to the above.
In order to handle this ’mild paracontrolled calculus’, one has, on a technical
level, to deal with integral operators and needs to understand how to
renormalize them correctly. The basic idea of this procedure will also be
introduced in my talk.
Finally, I will very briefly mention an alternative theory to handle singular