Two-dimensional semilinear stochastic heat equations with critical long-range correlations

Source abstract

We study a $(2+1)$-dimensional semilinear stochastic heat equation in which the noise is correlated like $f(\log |z|)/|z|^2$ as $|z|\to \infty$. Under an appropriate rescaling and logarithmic attenuation, we show that the limiting two-point statistics are given by the solution to a forward-backward stochastic differential equation. As shown in the work of Cosco, Cottini, and Donadini (2025), with whose results our study agrees in the linear case, the interesting feature of the two-dimensional, long-range problem is that one sees a varying influence on the limiting distribution from a range of spatial scales. We show that this phenomenon appears in the decoupling function for the limiting object and that the function $f$ modulates the weight of the contribution of each spatial scale.