Energy solutions to the stochastic Burgers equation were introduced by Goncalves-Jara to derive fluctuations for weakly asymmetric particle systems. They give a probabilistically weak formulation of this singular SPDE and thus provide a complementary point of view to the pathwise perspective of regularity structures. As observed by Gubinelli-Jara, the construction of energy solutions extends to many parabolic stochastic PDEs with Gaussian invariant measures and quadratic nonlinearities. More recently, in a joint work with Gubinelli we were able to prove weak uniqueness for energy solutions of the fractional multi-component stochastic Burgers equation, and in follow-up works by Gubinelli-Turra and Luo-Zhu this was extended to certain 2d fluid dynamic models. By taking a more abstract, functional analytic point of view, Lukas Gräfner was able to greatly simplify the proof of weak uniqueness and to turn it into a general result which applies to a wide range of equations and even proves well-posedness of some scaling critical equations. In my talk I will discuss Gräfner's functional analytic approach and present some examples including d-dimensional hyperviscous stochastic Navier-Stokes equations with white noise invariant measures.
