Superdiffusivity for Brownian Motion in a Poissonian potential with long range correlation I: Lower bound on the volume exponent

We study trajectories of d-dimensional Brownian Motion in Poissonian potential up to the hitting time of a distant hyperplane. Our Poissonian potential V is constructed from a field of traps whose centers location is given by a Poisson Point Process and whose radii are IID distributed with a common distribution that has unbounded support; it has the particularity of having long-range correlation. We focus on the case where the law of the trap radii v has power-law decay and prove that superdiffusivity hold under certain condition, and get a lower bound on the volume exponent. Results differ quite much with the one that have been obtained for the model with traps of bounded radii by Wuhtrich (Ann. Probab. 26 (1998) 1000-1015, Ann. Inst. Henri Poincare Probab. Stat. 34 (1998) 279-308): the superdiffusivity phenomenon is enhanced by the presence of correlation.