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The quenched asymptotics for nonlocal Schrödinger operators with Poissonian potentials

We study the quenched long time behaviour of the survival probability up to time $t$, $\mathbf{E}_x\big[e^{-\int_0^t V^ω(X_s){\rm d}s}\big],$ of a symmetric Lévy process with jumps, under a sufficiently regular Poissonian random potential $V^ω$ on $\mathbb{R}^d$. Such a function is a probabilistic solution to the parabolic eq. involving the nonlocal Schrödinger operator based on the generator of $(X_t)_{t \geq 0}$ with potential $V^ω$. For a large class of processes and potentials, we determine rate functions $η(t)$ and positive constants $C_1, C_2$ such that \[-C_1 \leq \liminf_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^ω(X_s){\rm d}s}\big]}{η(t)} \leq \limsup_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^ω(X_s){\rm d}s}\big]}{η(t)} \leq -C_2, \] almost surely with respect to $ω$, for every fixed $x \in \mathbb{R}^d$. The functions $η(t)$ and the bounds $C_1, C_2$ 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 $C_1=C_2$, i.e. the limit exists. Representative examples in this class are relativistic stable processes with Lévy-Khintchine exponents $ψ(ξ) = (|ξ|^2+m^{2/α})^{α/2}-m$, $α\in (0,2)$, $m>0$, for which \[\lim_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^ω(X_s)ds}\big]}{t/(\log t)^{2/d}} = \fracα{2} m^{1-\frac{2}α} \, \left(\frac{ρω_d}{d}\right)^{\frac{d}{2}} \, λ_1^{BM}(B(0,1)), \quad \mbox{for almost all $ω$,}\] where $λ_1^{BM}(B(0,1))$ is the principal eigenvalue of the Brownian motion in the unit ball, $ω_d$ is the Lebesgue measure of a unit ball and $ρ>0$ corresponds to $V^ω$. We also identify two interesting regime changes ('transitions') in the growth properties of $η(t)$