Project description
Stochastic nonlinear wave equations under study
Dispersive partial differential equations (PDEs) are fundamental in the understanding of various important physical phenomena in a wide array of scientific disciplines including telecommunications, plasma physics and non-linear optics. The EU-funded SingStocDispDyn project aims to develop novel mathematical tools to advance fundamental research at the interface between dispersive PDEs and stochastic analysis, by analysing concrete examples of challenging open problems. Building on a recent work on 3D stochastic nonlinear wave (NLW) equation with a quadratic non-linearity, the project will investigate the well-posed problem of the 3D stochastic cubic NLW equation with an additive space-time white noise. SingStocDispDyn's researchers will also work on pathwise solutions for dispersive PDEs with multiplicative noises, by combining the Fourier restriction norm method and the rough path theory.
Objective
This proposal is concerned with the study of singular stochastic dispersive partial differential equations (PDEs), broadly interpreted, with stochastic forcing and / or random initial data. This is a young emerging field, attracting more and more attention. In recent years, we have witnessed outstanding advances in the theory of singular stochastic parabolic PDEs. Our understanding of the dispersive counterpart is, however, much poorer. The main objective of this proposal is to develop novel mathematical ideas and tools and fundamentally advance our understanding of singular stochastic dispersive PDEs by working on concrete examples of challenging open problems.
Over the last ten years, there has been significant progress at the interface of dispersive PDEs and stochastic analysis and I have been one of the leading mathematicians in this development. In particular, my recent work on the three-dimensional stochastic nonlinear wave equation (NLW) with a quadratic nonlinearity has opened up new research horizons, which we will explore in this proposal.
1. We will investigate the well-posedness issue of the three-dimensional (damped) stochastic cubic NLW with an additive space-time white noise. The solution theory for the parabolic counterpart (the so-called stochastic quantisation equation) was settled by Hairer (2014). The corresponding question for the wave equation is one of the major open questions in this field. We will develop a paracontrolled approach to solve this challenging open problem. Moreover, we address other related problems of independent interest, including the two-dimensional hyperbolic sine-Gordon model, diffusive scaling limit of damped stochastic NLW and singular stochastic nonlinear Schrdinger dynamics.
2. We will also build a pathwise solution theory for stochastic dispersive PDEs with multiplicative noises by combining the Fourier restriction norm method and the rough path theory.
Fields of science (EuroSciVoc)
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
CORDIS classifies projects with EuroSciVoc, a multilingual taxonomy of fields of science, through a semi-automatic process based on NLP techniques. See: The European Science Vocabulary.
- natural sciences mathematics pure mathematics mathematical analysis differential equations partial differential equations
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Programme(s)
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Multi-annual funding programmes that define the EU’s priorities for research and innovation.
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H2020-EU.1.1. - EXCELLENT SCIENCE - European Research Council (ERC)
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Topic(s)
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Calls for proposals are divided into topics. A topic defines a specific subject or area for which applicants can submit proposals. The description of a topic comprises its specific scope and the expected impact of the funded project.
Funding Scheme
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Funding scheme (or “Type of Action”) inside a programme with common features. It specifies: the scope of what is funded; the reimbursement rate; specific evaluation criteria to qualify for funding; and the use of simplified forms of costs like lump sums.
ERC-COG - Consolidator Grant
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Call for proposal
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Procedure for inviting applicants to submit project proposals, with the aim of receiving EU funding.
(opens in new window) ERC-2019-COG
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EH8 9YL Edinburgh
United Kingdom
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