The quenched asymptotics for nonlocal Schrödinger operators with Poissonian potentials

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We study the quenched long time behaviour of the survival probability up to time t, Eₓ[e-∫₀ᵗ Vω(Xₛ) ds], of a symmetric Lévy process with jumps, under a sufficiently regular Poissonian random potential Vω on Rᵈ. Such a function is a probabilistic solution to the parabolic eq. involving the nonlocal Schrödinger operator based on the generator of (Xₜ)ₜ ≥ ₀ with potential Vω. For a large class of processes and potentials, we determine rate functions η(t) and positive constants C₁, C₂ such that -C₁ ≤ ₜ → ∞ Eₓ[e-∫₀ᵗ Vω(Xₛ) ds]η(t) ≤ ₜ → ∞ Eₓ[e-∫₀ᵗ Vω(Xₛ) ds]η(t) ≤ -C₂, almost surely with respect to ω, for every fixed x ∈ Rᵈ. The functions η(t) and the bounds C₁, C₂ heavily depend on the intensity of large jumps of the process. In particular, if its decay at infinity is `sufficiently fast', then we prove that C₁=C₂, i.e. the limit exists. Representative examples in this class are relativistic stable processes with Lévy-Khintchine exponents ψ(ξ) = (|ξ|²+m²/α)α/²-m, α∈ (0,2), m>0, for which ₜ → ∞ Eₓ[e-∫₀ᵗ Vω(Xₛ)ds]t/(t)²/ᵈ = α2 m1-2α (ρωd/d)ᵈ/² λ₁BM(B(0,1)), for almost all ω, where λ₁BM(B(0,1)) is the principal eigenvalue of the Brownian motion in the unit ball, ωd is the Lebesgue measure of a unit ball and ρ>0 corresponds to Vω. We also identify two interesting regime changes ('transitions') in the growth properties of η(t)