The quenched asymptotics for nonlocal Schrödinger operators with Poissonian potentials

Abstract

We study the quenched long time behaviour of the survival probability up to time $t$, ${\mathbf{E}}_x\left[e^{-{\int }_0^t{V}^{\omega }(X_s)\mathrm{d}s}\right],$ of a symmetric Lévy process with jumps, under a sufficiently regular Poissonian random potential $V^{\omega }$ 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 $(X_t{)}_{t\ge 0}$ with potential $V^{\omega }$. For a large class of processes and potentials, we determine rate functions $\eta (t)$ 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 $\omega$, for every fixed $x\in {\mathbb{R}}^d$. The functions $\eta (t)$ 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 $\psi (\xi )=(|\xi {|}^2+m^{2/\alpha }{)}^{\alpha /2}-m$, $\alpha \in (0,2)$, $m>0$, for which \[\lim_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^ω(X_s)ds}\big]}{t/(\log t)^{2/d}} = \fracα{2} m^{1-\frac{2}α} \, \left(\frac{ρω_d}{d}\right)^{\frac{d}{2}} \, λ_1^{BM}(B(0,1)), \quad \mbox{for almost all $\omega$,}\] where ${\lambda }_1^{BM}(B(0,1))$ is the principal eigenvalue of the Brownian motion in the unit ball, ${\omega }_d$ is the Lebesgue measure of a unit ball and $\rho >0$ corresponds to $V^{\omega }$. We also identify two interesting regime changes ('transitions') in the growth properties of $\eta (t)$

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