Stochastic PDEs and Renormalisation

The field of stochastic partial differential equations (SPDEs) has been revolutionised in the last decade by breakthrough works of Hairer, Gubinelli-Imkeller-Perkowski, and many others. A new understanding of renormalised solution theories emerged, solving long-standing singular equations arising in various areas of probability and mathematical physics. The purpose of this project is to study a number of important questions in the field, open new directions, and challenge central open problems:

(i) Launch the investigation of singular SPDEs that preserve Gibbs measures of distributional Hamiltonians such as the density of self-repellent polymers;

(ii) Tackle the question of a quasilinear renormalisation formula, the last remaining component of the quasilinear solution theory;

(iii) Develop an efficient quantitative approximation theory of singular SPDEs, removing the criticality barrier from the rate of convergence.