Brownian motion in a renormalized inverse-square Poisson potential
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Abstract
We study a d-dimensional Brownian motion W moving in a potential V based on a Poisson point process. It depends on the properties of the so called shape function K, whether the corresponding quenched and annealed Gibbs measures are well defined. In some cases, the lack of finiteness of the normalizing constant can be overcome by applying a renormalization introduced in [CK12]. Taking K(x) = θ|x|^(−p), the finiteness of the positive quenched exponential moments depends on whether p < 2 or p > 2. In the case p = 2, d = 3 we show that a phase transition occurs at θ = 1/16 which is closely related to the optimal constant in the classical Hardy inequality. With the help of a multipolar Hardy inequality we determine the asymptotic behaviour of the quenched exponentiel moment as t ↑ ∞.