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Singular stochastic PDEs. (English) Zbl 1373.60111

Jang, Sun Young (ed.) et al., Proceedings of the International Congress of Mathematicians (ICM 2014), Seoul, Korea, August 13–21, 2014. Vol. IV: Invited lectures. Seoul: KM Kyung Moon Sa (ISBN 978-89-6105-807-0/hbk; 978-89-6105-803-2/set). 49-73 (2014).
Summary: We present a series of recent results on the well-posedness of very singular parabolic stochastic partial differential equations. These equations are such that the question of what it even means to be a solution is highly non-trivial. This problem can be addressed within the framework of the recently developed theory of “regularity structures”, which allows to describe candidate solutions locally by a “jet”, but where the usual Taylor polynomials are replaced by a sequence of custom-built objects. In order to illustrate the theory, we focus on the particular example of the Kardar-Parisi-Zhang equation, a popular model for interface propagation.
For the entire collection see [Zbl 1314.00106].

MSC:

60H15 Stochastic partial differential equations (aspects of stochastic analysis)
81S20 Stochastic quantization
82C28 Dynamic renormalization group methods applied to problems in time-dependent statistical mechanics
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