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

Let ₜ≥ 0 be a d-dimensional supercritical super-Brownian motion started from the origin with branching mechanism ψ. Denote by Rₜ:=inf>0:Xₛ(∈ Rᵈ:|x|≥ r\)=0,~∀~0≤ s≤ t\ the radius of the minimal ball (centered at the origin) containing the range of ₛ≥ 0 up to time t. In {Pinsky}, Pinsky proved that condition on non-extinction, limt→∞Rₜ/t=√2β in probability, where β:=-ψ'(0). Afterwards, Engl\"{a}nder {Englander04} studied the lower deviation probabilities of Rₜ. For the upper deviation probabilities, he [Conjecture 8]{Englander04} conjectured that for ρ>√2β, limt→∞1/t(Rₜ≥ ρ t)=-(ρ²/2-β). In this note, we confirmed this conjecture.