Bi-parameter paracontrolled approach to singular stochastic wave equations

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Nonlinear dispersive partial differential equations (PDEs), such as the nonlinear wave equations (NLW), appear ubiquitously as models describing wave propagation in various branches of physics and engineering. In particular, it is well known that the one-dimensional wave equation describes the motion of a vibrating string. In a physical setting, such a vibrating string is susceptible to external forcing which is often random. Such a random external forcing is well approximated by a white noise in many situations. For this reason, it is of fundamental physical importance to study the stochastic NLW forced by space-time white noise. At the same time, such a problem also poses significant analytical challenges due to the irregularity of the space-time white noise. The main aim of this proposal is to advance our theoretical understanding of the one-dimensional stochastic NLW with multiplicative space-time white noise forcing by working on concrete examples of challenging open problems.The main difficulty of mathematical analysis on singular stochastic PDEs with white noise forcing comes from the irregularity of the white noise. Over the last decade, we have seen a tremendous progress in the study of singular stochastic PDEs with white noise forcing. In the parabolic setting, this development was led by a 2014 Fields medalist, Hairer (EPFL, Switzerland), and by Gubinelli (Oxford). Over the last five years, the principal investigator (PI) has made a substantial, world-leading contribution to the development of our theoretical understanding of singular stochastic NLW.In the proposed projects, the PI will study several important models of one-dimensional stochastic NLW with multiplicative space-time white noise forcing and aims to resolve challenging open problems by establishing their pathwise well-posedness. The PI plans to achieve this goal by developing an entirely new analytical framework which allows him to handle singular multiplicative noises in a pathwise manner.