[Paper Review] Second order asymptotics for Brownian motion in a heavy tailed Poissonian potential

Ryoki Fukushima|arXiv (Cornell University)|Oct 19, 2010

Spectral Theory in Mathematical Physics12 references16 citations

TL;DR

This paper establishes second-order asymptotics for the Feynman-Kac functional of Brownian motion in a heavy-tailed Poissonian potential with power-law decay $|x|^{-eta}$, where $d < \beta < d+2$. Using variational methods and large deviation techniques, it derives precise moment and almost sure asymptotics up to the second order, including the second-order correction to the survival probability and the integrated density of states of the associated random Schrödinger operator.

ABSTRACT

We consider the Feynman-Kac functional associated with a Brownian motion in a random potential. The potential is defined by attaching a heavy tailed positive potential around the Poisson point process. This model was first considered by Pastur (1977) and the first order term of the moment asymptotics was determined. In this paper, both moment and almost sure asymptotics are determined up to the second order. As an application, we also derive the second order asymptotics of the integrated density of states of the corresponding random Schrödinger operator.

Motivation & Objective

• To derive second-order asymptotics for the Feynman-Kac functional of Brownian motion in a heavy-tailed Poissonian potential.
• To extend Pastur's first-order moment asymptotics to include second-order corrections.
• To establish almost sure asymptotics of the survival probability up to the second order.
• To determine the second-order asymptotics of the integrated density of states for the corresponding random Schrödinger operator.

Proposed method

• Use of variational methods to analyze the principal eigenvalue in large boxes.
• Application of large deviation techniques to control the tail behavior of the potential.
• Employment of the Feynman-Kac formula to connect the survival probability to the moment generating function.
• Use of Poisson point process moment generating functions (via characteristic functions and cumulants) to compute variance and higher moments.
• Taylor expansion and localization techniques to estimate the difference between $V_\omega(x)$ and $V_\omega(0)$ in the relevant region.
• Reduction of the integrated density of states problem to eigenvalue estimates in finite boxes using spectral comparison.

Experimental results

Research questions

• RQ1 What is the second-order correction to the moment asymptotics of the Feynman-Kac functional in the heavy-tailed Poissonian potential?
• RQ2 How do the typical paths of the Brownian motion behave in the heavy-tailed potential, and what is the almost sure asymptotic behavior of the survival probability?
• RQ3 What is the second-order asymptotic behavior of the integrated density of states for the random Schrödinger operator with such a potential?
• RQ4 How does the variance of the potential contribution affect the second-order term in the moment asymptotics?

Key findings

• The second-order asymptotics of the moment of the Feynman-Kac functional are derived as $\exp\left\{ -a_1 t^{d/\beta} - a_2 t^{d/\beta} (\log t)^{-\gamma} + o(t^{d/\beta} (\log t)^{-\gamma}) \right\}$ for some explicit constants $a_1, a_2, \gamma$.
• The almost sure asymptotics of the survival probability are shown to match the moment asymptotics up to the second order, confirming typical path behavior.
• The second-order correction to the integrated density of states is derived via eigenvalue estimates in large boxes, showing a $\log t$-dependent correction term.
• The variance of the potential contribution is shown to be negligible in the second-order term due to localization and decay properties of $|x|^{-\beta}$.
• The principal eigenvalue in a large box $(-t,t)^d$ is shown to be bounded above by a term decaying as $ (\log t)^{-(\beta - d)/d} $, which controls the integrated density of states.