Mathematics

Stochastic partial differential equations (SPDEs) naturally arise in the mathematical description of systems influenced by randomness, such as fluctuating interfaces, turbulent fluids, or fields in statistical mechanics and quantum field theory. While many SPDEs can be analyzed using classical techniques, a particularly challenging class—singular SPDEs—has become a central topic in modern probability theory and mathematical physics. These equations are singular because their solutions are too irregular for the nonlinear terms to be well-defined in the classical sense. This phenomenon parallels the ultraviolet divergences in quantum field theory (QFT), where renormalization is needed to define interacting models.

 

The first part of the course will introduce and motivate singular SPDEs, explain the origin of their singularities, and discuss the necessity of renormalization in constructing their solutions. We will review core examples, including the KPZ and stochastic quantization equations, emphasizing their links to quantum field theory and to scaling limits of microscopic models.

 

The second part will focus on the flow equation approach, a robust framework applicable to a broad class of singular SPDEs—including those with fractional Laplacians—throughout the entire subcritical regime. Inspired by Wilson’s renormalization group, this method studies the coarse-grained process, which captures the behavior of solutions across spatial scales. The corresponding flow equation describes how the nonlinear terms in the effective dynamics evolve with the coarse-graining scale, playing a role analogous to the Polchinski equation in QFT. The renormalization problem is then solved inductively by imposing appropriate boundary conditions on the flow equation.