Variance renormalisation of singular SPDEs

Scaling arguments give a natural guess at the regularity condition on the noise in a stochastic PDE for a local solution theory to be possible, using the machinery of regularity structures or paracontrolled distributions. This guess of ``subcriticality'' is often, but not always, correct. In cases when it is not, a the blowup of the variance of certain nonlinear functionals of the noise necessitates a different, multiplicative renormalisation. This led to a general prediction and the first results in the case of the KPZ equation in [Hairer '24]. We discuss recent developments towards confirming this prediction. Based on joint works with Fabio Toninelli and Yueh-Sheng Hsu.