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Let \Xₜ\ₓ ₀ be a d-dimensional supercritical super-Brownian motion started from the origin with branching mechanism. Denote by Rₜ: =\r>0: Xₛ (\{x Rᵈ: |x| r\) =0, ~~0 s t\} the radius of the minimal ball (centered at the origin) containing the range of \Xₛ\ₒ ₀ up to time t. In Pinsky, Pinsky proved that condition on non-extinction, ₓRₜ/t=2 in probability, where: =-' (0). Afterwards, Engl\"ander Englander04 studied the lower deviation probabilities of Rₜ. For the upper deviation probabilities, he 8Englander04 conjectured that for > 2, ₓ1t (Rₜ t) =- (²2-). In this note, we confirmed this conjecture.
Shuxiong Zhang (Thu,) studied this question.