Most techniques for solving singular SPDEs, such as regularity structures, are based on pathwise calculus. It would be interesting to study singular SPDEs from a more probabilistic perspective, for example via the martingale problem. In general, that is a too difficult problem at the moment, but there are some equations for which we can do this. I will explain the ideas on the example of the conservative stochastic Burgers equation and indicate how to extend the results to a larger class of equations that share a similar structure. A novelty of this approach is that it allows to prove (weak) well-posedness for some scaling critical singular SPDEs.

Based on works with Massimiliano Gubinelli and Lukas Gräfner.