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We study the maximal displacement and related population for a branchingBrownian motion in Euclidean space in terms of the principal eigenvalue of anassociated Schr\"odinger type operator. We first determine their growth rateson the survival event. We then establish the upper deviation for the maximaldisplacement under the possibility of extinction. Under the non-extinctioncondition, we further discuss the decay rate of the upper deviation probabilityand the population growth at the critical phase.

Maximal displacement and population growth for branching Brownian motions