Project description
Semi-linear stochastic partial differential equations: global solutions’ behaviours
Partial differential equations are fundamental to describing processes in which one variable is dependent on two or more others – most situations in real life. Stochastic partial differential equations (SPDEs) describe physical systems subject to random effects. In the description of scaling limits of interacting particle systems and in quantum field theories analysis, the randomness is due to fluctuations related to noise terms on all length scales. The presence of a non-linear term can lead to divergencies. Funded by the European Research Council, the GE4SPDE project will describe the global behaviour of solutions of some of the most prominent examples of semi-linear SPDEs, building on the systematic treatment of the renormalisation procedure used to deal with these divergencies.
Objective
"The project is concerned with the global behaviour of solutions to Stochastic Partial Differential Equations (SPDEs) from Mathematical Physics which arise e.g. in the description of scaling limits of interacting particle systems and in the analysis of Quantum Field Theories. The equations contain noise terms which describe random fluctuations and act on all length scales. In this situation the presence of a non-linear term can lead to divergencies. A subtle renormalisation procedure, which amounts to removing infinite terms, is needed. Over the last years the understanding of non-linear SPDEs has been revolutionised and a systematic treatment of the renormalisation procedure has been achieved. This led to a short-time well-posedness theory on compact domains for a large class of highly relevant semi-linear SPDEs. In this project, I will describe the global - both in time and over infinite domains - behaviour of solutions of some of the most prominent examples, by combining PDE techniques for the non-linear equations without noise and the improved understanding of the subtle small-scale stochastic cancellations. I have already pioneered such a programme in an important special case, the dynamic Phi-4 model. The project has three specific strands: A) Proving estimates for the stochastic quantisation equations of the Sine-Gordon and Liouville Quantum Gravity models and eventually Gauge theories, and to giving a PDE-based approach to the celebrated 1-2-3 scaling of the KPZ equation, B) giving PDE-based constructions of Phi-4 models in fractional dimension and describing phase transitions in terms of mixing properties of the dynamics, C) treating degenerate parabolic equations and exploring if systems that fail to satisfy a fundamental ""sub-criticality"" scaling assumption can still be treated using SPDE techniques."
Fields of science
- natural sciencesmathematicsapplied mathematicsmathematical physics
- natural sciencesphysical sciencesquantum physicsquantum field theory
- natural sciencesmathematicspure mathematicsmathematical analysisdifferential equationspartial differential equations
- natural sciencesphysical sciencestheoretical physics
Keywords
Programme(s)
- HORIZON.1.1 - European Research Council (ERC) Main Programme
Funding Scheme
HORIZON-ERC - HORIZON ERC GrantsHost institution
48149 MUENSTER
Germany