Asymptotics and the sub-limit at $L^{2}$-criticality of higher moments for the SHE in dimension $d\geq 3$

In this article, we consider the $d$-dimensional mollified stochastic heat equation (SHE) when the mollification parameter is turned off. Here, we concentrate on the high-dimensional case $d \geq 3$. Recently, the limiting higher moments of the two-dimensional mollified SHE have been established. However, this problem in high dimensions remains unexplored to date. The main theorems of this article aim to answer this question and prove some related properties: (1) Our first main result, based on the spectral theorem for the unbounded operator, proves the divergence of the higher moments of the high-dimensional mollified SHE even when the system is strictly inside the $L^{2}$-regime. This phenomenon is completely opposite to its two-dimensional counterpart; (2) To further differentiate the nature of the high-dimensional case from the case in two dimensions, our second main result proves the unboundedness of the sub-limiting higher moments of the three-dimensional mollified SHE at the $L^{2}$-criticality. Here, the sub-limiting higher moment is a natural limit of the higher moment of the three-dimensional mollified SHE at the $L^{2}$-criticality; (3) As an application, we provide partial results for the conjecture about the high-order critical regimes of the continuous directed polymer. The other byproduct of the above results gives a proper estimate for the critical exponent of the polymer in the $L^{2}$-regime.