Spatial Brownian motion in renormalized Poisson potential: A critical case

الأصل · EN

Let Bₛ be a three dimensional Brownian motion and ω(dx) be an independent Poisson field on R³. It is proved that for any t>0, conditionally on ω(·), * E₀ {θ∫₀ᵗ V(Bₛ) ds} < ∞ a.s. & if θ< 1/16, = ∞ a.s. & if θ> 1/16, where V(x) is the renormalized Poisson potential V(x)=∫ᵣ₃ 1/| x-y |² [ω(dy)-dy]. Then the long term behavior of the quenched exponential moment * is determined for θ∈ (0, 1/16) in the form of integral tests. This paper exhibits and builds upon the interrelation between the exponential moment * and the celebrated Hardy's inequality ∫ᵣ₃ f²(x)/| x |² dx ≤ 4 ∇ f₂², 2in f ∈ W¹,²(R³).