The multiplicative-noise correction conjecture for stochastic Burgers discretisations

Let $g$ be as in the scalar stochastic Burgers equation and let $f$ be a smooth bounded function with bounded derivatives of all orders. Consider the regularised equation

$$du=\nu\,\partial_x^2u\,dt+g(u)D_{\varepsilon}u\,dt+f(u)\,dw.$$

Multiplicative-noise correction conjecture. As $\varepsilon\to0$, its solution converges to the solution of

$$du=\nu\,\partial_x^2u\,dt+g(u)\,\partial_xu\,dt-\frac{1}{4\nu}g'(u)f^2(u)\,dt+f(u)\,dw.$$

The correction is motivated by the expected local quadratic variation, proportional to $f^2(u)$; the source presents the result as an expectation and does not give a proof.