Fluctuations of she and kpz equations with non-integrable spatial correlations

This thesis examines the large scales fluctuations of the solution to a nonlinear stochastic heat equation (SHE) and the KPZ equation in dimensions $d > 3$, in the presence of long-range spatial correlations. Specifically, the equations are driven by Gaussian noise that is white in time but has non-integrable spatial covariance (at infinity), with polynomial decay rate $\kappa \in (2, d)$. We prove that the fluctuations of both equations, when appropriately rescaled, converge to a Gaussian limit described by the corresponding additive stochastic heat equation. Unlike the case of compactly supported covariance, the noise in the limit equation retains spatial correlation with a Riesz kernel $|x|^{-\kappa}$ as covariance, along with a different effective variance and fluctuations scales. Additionally, by considering the spaces of locally Hölder continuous distributions $\C^\alpha$, we demonstrate convergence in optimal Hölder topologies as distribution-valued processes. To establish our main results, we also prove a priori moment bounds on the solutions and their Malliavin derivatives, as well as the existence of space-time stationary solutions, which underpin the fluctuations theory of both models. Interestingly, in contrast to the case of integrable covariance, we show that the KPZ equation converges in probability to its scaling limit.