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 (Ren et al. in Stoch Proc Appl 137:1–34, 2021), we showed that, under some conditions, M_t-m(t) converges in distribution to a randomly shifted Gumbel random variable, where m(t)=c_0t-c_1log t . In the same paper, we also studied the upper large deviation of M_t , i.e., the asymptotic behavior of ℙ (M_t>δ c_0t) for δ≥ 1 . In this paper, we study the lower large deviation of M_t , i.e., the asymptotic behavior of ℙ (M_t≤δ c_0t|𝒮) for δ <1 , where 𝒮 is the survival event.