Massimiliano Gubinelli and Nicolas Perkowski
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.
Massimiliano Gubinelli and Nicolas Perkowski. Probabilistic approach to the stochastic Burgers equation. Papers of Massimiliano Gubinelli, 2017, version 1. https://mgubi.org/papers/paper/burgers-probabilistic/
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