Universality and long-time behavior of quasilinear singular stochastic partial differential equations

• Applicant: Dr. Markus Tempelmayr
• Subject Area: Mathematics
• Term: since 2024
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 552900305

Singular stochastic PDEs (SPDE) have attracted considerable attention in the past decade both from an applied as well as from a pure mathematics community: they often arise as "universal" objects in physical models, and their study is mathematically challenging, requiring tools from algebra, analysis, and probability. While much focus has been put on equations connecting to (Euclidean) quantum field theory, models from mesoscopic physics have been somewhat neglected; a prime example is the thin-film equation with thermal noise. As a contribution towards filling this gap I propose to study a certain class of quasilinear singular SPDEs, in particular investigating their (weak) universality properties and their long-time behavior. More precisely I propose to build upon the approach to singular SPDEs recently developed in collaboration with Linares, Otto, and Tsatsoulis, and by Sauer and Smith, toi) capture logarithms in the scaling of a term-by-term description of solutions and understand the degree of freedom in renormalization that comes along,ii) prove weak universality results using Malliavin calculus,iii) prove global (in time) existence of solutions of the thin-film equation with thermal noise and invariance of the Gaussian free field conditioned to positive functions.

• DFG Programme: WBP Fellowship
• International Connection: Switzerland
• Host: Professor Dr. Martin Hairer