Upper deviation probabilities for the range of a supercritical super-Brownian motion

Abstract

Let $\{X_t{\}}_{t\ge 0}$ be a $d$-dimensional supercritical super-Brownian motion started from the origin with branching mechanism $\ddot{I}ˆ$. Denote by $R_t:=\inf \{r>0:X_s(\{x\in {\mathbb{R}}^d:|x|\ge r\})=0,\forall 0\le s\le t\}$ the radius of the minimal ball (centered at the origin) containing the range of $\{X_s{\}}_{s\ge 0}$ up to time $t$. In \cite{Pinsky}, Pinsky proved that condition on non-extinction, ${\lim }_{t\to \infty }{R}_t/t=\sqrt{2{\hat{I}}^2}$ in probability, where ${\hat{I}}^2:=-\ddot{I}{ˆ}^{\prime }(0)$. Afterwards, Engländer \cite{Englander04} studied the lower deviation probabilities of $R_t$. For the upper deviation probabilities, he \cite[Conjecture 8]{Englander04} conjectured that for $\ddot{I}>\sqrt{2{\hat{I}}^2}$, $$\underset{t\to \infty }{\lim }\frac{1}{t}\log \mathbb{P}(R_t\ge \ddot{I}t)=-\left(\frac{\ddot{I}{}^2}{2}-{\hat{I}}^2\right).$$ In this note, we confirmed this conjecture.

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