WIAS Preprint No. 2482, (2018)

Brownian motion in attenuated or renormalized inverse-square Poisson potential



Authors

  • Nelson, Peter
  • Soares dos Santos, Renato

2010 Mathematics Subject Classification

  • 60J65 60G55 60K37 35J10 35P15

Keywords

  • Brownian motion in Poisson potential, parabolic Anderson model, inverse square potential, multipolar Hardy inequality

DOI

10.20347/WIAS.PREPRINT.2482

Abstract

We 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].

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