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- ItemBrownian motion in attenuated or renormalized inverse-square Poisson potential(Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2018) Nelson, Peter; Santos, Renato Soares dosWe consider the parabolic Anderson problem with random potentials having inverse-square singularities around the points of a standard Poisson point process in ℝ d, d ≥3. The potentials we consider are obtained via superposition of translations over the points of the Poisson point process of a kernel 𝔎 behaving as 𝔎 (x)≈ Θ x -2 near the origin, where Θ ∈(0,(d-2)2/16]. In order to make sense of the corresponding path integrals, we require the potential to be either attenuated (meaning that 𝔎 is integrable at infinity) or, when d=3, renormalized, as introduced by Chen and Kulik in [8]. Our main results include existence and large-time asymptotics of non-negative solutions via Feynman-Kac representation. In particular, we settle for the renormalized potential in d=3 the problem with critical parameter Θ = 1/16, left open by Chen and Rosinski in [9].