In this talk, I will review some of the recent results on the Stochastic Heat Equation (SHE) with multiplicative white noise in dimension d=2. The SHE is a stochastic PDE which is ill-defined in its critical dimension d=2 : in that case, very recent results show that a subtle normalisation procedure is needed to make sense of it.
I will present the probabilistic approach to this normalisation procedure, followed by Caravenna, Sun, Zygouras : it is based on the study of the directed polymer model, a statistical mechanics model which can be seen as a discretised version of the SHE. In a very specific critical window for the parameters, the model possess a non-trivial scaling limit, that Caravenna, Sun, Zygouras called Critical 2D Stochastic Heat Flow, and can be interpreted as a (notion of a) solution to the 2D SHE.
I will then review some of the properties of this Stochastic Heat Flow and present some of the results based on a joint work with F. Caravenna and N. Turchi.