Lower deviation for the supremum of the support of super-Brownian motion

We study the asymptotic behavior of the supremum $M_t$ of the support of a supercritical super-Brownian motion. In our recent paper (Stoch. Proc. Appl. 137 (2021), 1-34), we showed that, under some conditions, $M_t-m(t)$ converges in distribution to a randomly shifted Gumbel random variable, where $m(t)=c_{0}t-c_1\log t$. In the same paper, we also studied the upper large deviation of $M_t$, i.e., the asymptotic behavior of $\mathbb{P}(M_t>\delta c_{0}t)$ for $\delta \ge 1$. In this paper, we study the lower large deviation of $M_t$, i.e., the asymptotic behavior of $\mathbb{P}(M_t\le \delta c_{0}t\mathrm{\mid }\mathcal{S})$ for $\delta <1$, where $\mathcal{S}$ is the survival event.