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This paper continues a study on trajectories of Brownian Motion in a field ofsoft trap whose radius distribution is unbounded. We show here for bothpoint-to-point and point-to-plane model the volume exponent (the exponentassociated to transversal fluctuation of the trajectories) is strictly lessthan one and give an explicit upper bound that depends on the parameters of theproblem. In some specific cases, this upper bound matches the lower boundproved in the first part of this work and we get the exact value of the volumeexponent.

Superdiffusivity for Brownian motion in a Poissonian potential with long range correlation II: upper bound on the volume exponent