The quenched asymptotics for nonlocal Schr\"odinger operators with Poissonian potentials

Abstract

We study the quenched long time behaviour of the survival probability up to time $t$, $\mathbf{E}_x\big[e^{-\int_0^t V^{\omega}(X_s){\rm d}s}\big],$ of a symmetric L\'evy 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 Schr\"odinger operator based on the generator of $(X_t)_{t \geq 0}$ 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 \[-C_1 \leq \liminf_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^{\omega}(X_s){\rm d}s}\big]}{\eta(t)} \leq \limsup_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^{\omega}(X_s){\rm d}s}\big]}{\eta(t)} \leq -C_2, \] 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 L\'evy-Khintchine exponents $\psi(\xi) = (|\xi|^2+m^{2/\alpha})^{\alpha/2}-m$, $\alpha \in (0,2)$, $m>0$, for which \[\lim_{t \to \infty} \frac{\log \mathbf{E}_x\big[{\rm e}^{-\int_0^t V^{\omega}(X_s)ds}\big]}{t/(\log t)^{2/d}} = \frac{\alpha}{2} m^{1-\frac{2}{\alpha}} \, \left(\frac{\rho \omega_d}{d}\right)^{\frac{d}{2}} \, \lambda_1^{BM}(B(0,1)), \quad \mbox{for almost all $\omega$,}\] 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 $\eta(t)$