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