On the variance blow-up phenomenon of singular SPDEs
Vienna Probability Seminar
For singular SPDEs, solutions are typically given by the limit of appropriately renormalised random PDEs driven by mollified noises. Under the assumption of subcriticality, the black-box theory of regularity structures provides a systematic way of renormalisation. However, subcriticality alone does not promise the convergence of such renormalised solution, as stochastic objects required to make sense of the solution theory can have exploding variance when the mollification is removed. We will look at two examples which exhibit this phenomenon. In the first case, we will consider a supercritical SPDE and introduce an additional renormalisation in a multiplicative way, which leads to a limit different from what one would heuristically guess. In the second case, we start from a subcritical SPDE and consider the fluctuation of the renormalised solution around its limit; in particular, we prove a central limit-type theorem which characterises the fluctuation scale as
well as its limiting profile in law, which again solves a singular SPDE.
This talk is based on joint works with Mt Gerencsr and Rhys Steele.