A forward-backward SDE from the 2D nonlinear stochastic heat equation

We consider a nonlinear stochastic heat equation in spatial dimension $d=2$, forced by white-in-time multiplicative Gaussian noise with correlation length $\varepsilon>0$ but divided factor of $\sqrt{\log\varepsilon^{-1}}$. impose condition on the Lipschitz constant nonlinearity so that problem is "weak noise" regime. show that, as $\varepsilon\downarrow0$, one-point distribution solution converges, limit characterized terms to forward-backward differential (FBSDE). also characterize limiting multipoint statistics solution, when points are chosen appropriate scales, similar terms. Our approach new even for linear case, which FBSDE can be solved explicitly and we recover results Caravenna, Sun, Zygouras (Ann. Appl. Probab. 27(5):3050--3112, 2017).