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

Let $\{X_t{\}}_{t\ge 0}$ be a $d$-dimensional supercritical super-Brownian motion started from the origin with branching mechanism $\psi$. Denote by $R_t:=\inf \{r>0:X_s(\{x\in {\mathbb{R}}^d:\mathrm{\mid }x\mathrm{\mid }\ge r\})=0,\mathrm{\forall }0\le s\le t\}$ 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, ${\lim }_{t\to \mathrm{\infty }}{R}_t\mathrm{/}t=\sqrt{2\beta }$ in probability, where $\beta :=-{\psi }^{\mathrm{\prime }}(0)$. 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 $\rho >\sqrt{2\beta }$, $$\underset{t\to \mathrm{\infty }}{\lim }\frac{1}{t}\log \mathbb{P}(R_t\ge \rho t)=-(\frac{{\rho }^2}{2}-\beta )\mathrm{.}$$ In this note, we confirmed this conjecture.