On the space-time fluctuations of the SHE and KPZ equation in the entire $L^{2}$-regime for spatial dimensions $d \geq 3$

Abstract: In this article, we consider the mollified stochastic heat equation with flat initial data and the corresponding Kardar-Parisi-Zhang (KPZ) equation in dimension $d \geq 3$ when the mollification parameter is turned off. In addition, we assume that the strength of the space-time white noise is small enough, such that the solution of the mollified stochastic heat equation has a finite limiting second moment. In this case, with regard to the mollified KPZ equation, we show that the fluctuation of the mollified KPZ equation about its stationary solution converges in the sense of finite dimensional distributions towards a Gaussian free field. On the other hand, about the mollified stochastic heat equation, we show that the fluctuation of the mollified stochastic heat equation about its stationary solution converges in the sense of finite dimensional convergence towards the product of an i.i.d random field and an independent Gaussian free field. Here the i.i.d random field is generated by the limiting partition function of the continuous directed polymer. As an application, we prove the limiting fluctuations of the partition function and the free energy of the continuous directed polymer.