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 ψ. 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 [8], Pinsky proved that condition on non-extinction, ${lim}_{t\to \mathrm{\infty }}{R}_t∕t=\sqrt{2\mathit{\beta }}$ in probability, where $\mathit{\beta }:=-{\mathit{\psi }}^{\prime }(0)$. Afterwards, Engländer [1] studied the lower deviation probabilities of $R_t$. For the upper deviation probabilities, Engländer [1, Conjecture 8] conjectured that for $\mathit{\rho }>\sqrt{2\mathit{\beta }}$,

$$\underset{t\to \mathrm{\infty }}{lim}\frac{1}{t}log\mathbb{P}(R_t\ge \mathit{\rho }t|X_s({\mathbb{R}}^d)>0,\forall s>0)=-\left(\frac{{\mathit{\rho }}^2}{2}-\mathit{\beta }\right).$$

In this note, we confirmed this conjecture.