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_varepsilonu,dt+f(u),dw. Multiplicative-noise correction conjecture. As varepsilon→0, its solution converges to the solution of du=nu,partial_x^2u,dt+g(u),partial_xu,dt-frac14 nug'(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.

The multiplicative-noise correction conjecture for stochastic Burgers discretisations

The conjecture asserts that the regularised stochastic Burgers equation converges as $\varepsilon \to 0$ to an equation with the additional drift $-\frac{1}{4\nu}g'(u)f^2(u)$ . The catalogue source presents this as an expectation and reports no resolution.

Known results

Hairer and Maas, 2012: spatial discretisations with additive white noise produce a rigorous correction $-\frac{1}{4\nu}\Delta G(u)$ , but not the stated multiplicative-noise formula.
“Rough Burgers-like equations with multiplicative noise,” 2010: proves convergence for related approximations and identifies a reaction term involving $g'(u)\theta(u)^2$ , without establishing the exact coefficient here.
“Approximating rough stochastic PDEs,” 2012: proves convergence for broad spatial approximations, while emphasizing scheme-dependent limiting corrections.

Current status (as of September 2026): the exact multiplicative-noise correction conjecture remains open; related additive-noise and rough-equation results do not prove it.

The multiplicative-noise correction conjecture for stochastic Burgers discretisations

Primary source

Progress summary

Solutions 0