Paracontrolled approach to the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity

By Massimiliano Gubinelli, Herbert Koch, Tadahiro Oh. Journal of the European Mathematical Society (Print), 2018

76 citations recorded by the source corpus.

Using ideas from paracontrolled calculus, we prove local well-posedness of a renormalized version of the three-dimensional stochastic nonlinear wave equation with quadratic nonlinearity forced by an additive space-time white noise on a periodic domain. There are two new ingredients as compared to the parabolic setting. (i) In constructing stochastic objects, we have to carefully exploit dispersion at a multilinear level. (ii) We introduce novel random operators and leverage their regularity to overcome the lack of smoothing of usual paradifferential commutators.

Subfields: Control and Systems Engineering, Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/739b538816ab6770668e209090eee0ec5b98df55