tochastic heat equation plays a key role to construct a Cole-Hopf solution of KPZ equation in one dimensional case. It is natural to apply this idea to higher dimensional KPZ equation. However, stochastic heat equations in higher dimensions are ill posed.
In $2d$ case, Caravenna-Sun-Zygouras obtained a non-trivial limit ($2d$ critical stochastic heat flow, $2d$ critical SHF) of mollified stochastic heat equations with suitable renormalization in finite time dimensional sense.
In this talk, we discuss the tightness of the process and the martingale problem associated with $2d$ critical SHF via using an idea of measure valued processes.