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

TL;DR

This paper investigates the long-term behavior of the survival probability of jump processes in Poissonian potentials, establishing precise asymptotics and phase transitions depending on jump intensity and process decay properties.

Contribution

It provides the first detailed quenched asymptotic analysis for nonlocal Schrödinger operators with Poissonian potentials, including explicit rate functions and identification of phase transitions.

Findings

Asymptotic rate functions for survival probabilities are derived.

Existence of limits depends on the decay rate of the process at infinity.

Explicit results for relativistic stable processes are obtained.

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 $\eta(t)$