Quenched path limits and periodization stability for tilted Brownian motion in Poissonian potentials on $\mathbb{H}^d$

TL;DR

This paper investigates the existence and approximation of Q-processes for tilted Brownian motion in hyperbolic spaces with Poissonian potentials, establishing conditions under which these processes exist and can be approximated by periodic potentials.

Contribution

It demonstrates the existence of Q-processes for stationary random potentials on hyperbolic spaces under certain spectral and norm bounds, and introduces a method to approximate these processes using periodic potentials.

Findings

Q-processes exist almost surely for certain stationary random potentials.

The method extends to Poissonian potentials up to a specific sup norm bound.

Global ground states on foliated spaces serve as substitutes for non-existent L^2 ground states.

Abstract

We analyze the existence of Brownian motion tilted by a potential of full support on hyperbolic spaces $\mathbb{H}^d$. On compact spaces, it is classical that these path limits, called Q-processes, exist and can be directly defined using the ground state of the corresponding Schr\"odinger operator. On non-compact spaces like $\mathbb{H}^d$, the existence fails in general. We show that for \emph{stationary random} potentials on $\mathbb{H}^d$ with suitable spectral and sup norm bounds, the Q-processes exist a.s. For potentials that are factors of a Poisson point process, the method works up to sup norm $(d-1)^2/8$. In this case, we also show that the path limit can be approximated by periodic potentials. As a tool, we use the foliated space defined by the point process. It turns out that the global ground state of this foliated space serves as a substitute for the non-existing $L^2$ ground states on the leaves of the foliation. Restricting the global ground state to a leaf gives a generalized eigenwave that can be plugged into the usual machinery to get the Q-process.