Probabilistic Approach to the Stochastic Burgers Equation

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 (M. Gubinelli and N. Perkowski, Energy solutions of KPZ are unique. 2015, [18]), but we mainly emphasize the heuristic derivation and also we include a (very simple) extension of (M. Gubinelli and N. Perkowski, Energy solutions of KPZ are unique. 2015, [18]) to a non-stationary regime.