  We review the formulation of the stochastic Burgers equation as a martingale
problem. One way of understanding the difficulty in making sense of the
equation is to note that it is a stochastic PDE with distributional drift, so
we first review how to construct finite-dimensional diffusions with
distributional drift. We then present the uniqueness result for the stationary
martingale problem of [Gubinelli, Perkowski 2016], but we mainly emphasize the
heuristic derivation and also we include a (very simple) extension of
[Gubinelli, Perkowski 2016] to a non-stationary regime.

Probability