The Quenched Asymptotics for Nonlocal Schrödinger Operators with Poissonian Potentials
Kamil Kaleta, Katarzyna Pietruska-Pałuba · Potential Analysis · 2018
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 Vω on $\mathbb {R}^{d}$. Such a function is a probabilistic solution to the parabolic equation involving the nonlocal Schrödinger operator based on the generator of the process (Xt)t≥ 0 with potential Vω. For a large class of processes and potentials of finite range, we determine rate functions η(t) and compute explicitly the positive constants C1,C2 such that $ -C_{1} \leq \liminf \limits _{t \to \infty } \frac {\log \mathbf {E}_{x}\left [\mathrm {e}^{-{{\int }_{0}^{t}} V^{\omega }(X_{s})\mathrm {d}s}\right ]}{\eta (t)} \leq \limsup \limits _{t \to \infty } \frac {\log \mathbf {E}_{x}\left [\mathrm {e}^{-{{\int }_{0}^{t}} V^{\omega }(X_{s})\mathrm {d}s}\right ]}{\eta (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 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 we obtain that $\lim \limits _{t \to \infty } \frac {\log \mathbf {E}_{x}\left [\mathrm {e}^{-{{\int }_{0}^{t}} V^{\omega }(X_{s})ds}\right ]}{t/(\log t)^{2/d}} = \frac {\alpha }{2}m^{1-\frac {2}{\alpha }} \left (\frac {\rho \omega _{d}}{d}\right )^{\frac {d}{2}} \lambda _{1}^{BM}(B(0,1)), \quad \text {for almost all} \omega ,$ where $\lambda _{1}^{BM}(B(0,1))$ is the principal eigenvalue of the Brownian motion killed on leaving 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 the rates η(t) as the intensity of large jumps of the processes varies from polynomial to higher order, and eventually to stretched exponential order.