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

Miklos Abert, Adam Arras, Jaelin Kim

TL;DR

This work establishes the existence of quenched $Q$-processes for Brownian motion tilted by nonnegative potentials on ${\mathbb{H}^d}$, focusing on stationary random potentials and Poisson-factor potentials with a sup-norm bound. The authors develop a foliated-space framework in which a global ground state on the foliation substitutes for the leafwise $L^2$ ground states, enabling a Doob $h$-transform to define the limiting diffusion. They show that Poisson-factor potentials yield a positive lower bound on spectral gaps and that the associated $Q$-processes can be approximated by periodic potentials; this provides a robust stability result under Benjamini–Schramm convergence and connects randomness with periodization via the foliation. The results extend to BS limits of expanders and culminate in a periodic approximation theorem for the limit $Q$-process, while also outlining open questions for hard obstacle models. Overall, the paper links spectral theory on foliated spaces, hyperbolic geometry, and random media to yield rigorous existence and approximation results for tilted Brownian motion in negatively curved spaces.

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 Schrödinger 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.

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

TL;DR

This work establishes the existence of quenched -processes for Brownian motion tilted by nonnegative potentials on , focusing on stationary random potentials and Poisson-factor potentials with a sup-norm bound. The authors develop a foliated-space framework in which a global ground state on the foliation substitutes for the leafwise ground states, enabling a Doob -transform to define the limiting diffusion. They show that Poisson-factor potentials yield a positive lower bound on spectral gaps and that the associated -processes can be approximated by periodic potentials; this provides a robust stability result under Benjamini–Schramm convergence and connects randomness with periodization via the foliation. The results extend to BS limits of expanders and culminate in a periodic approximation theorem for the limit -process, while also outlining open questions for hard obstacle models. Overall, the paper links spectral theory on foliated spaces, hyperbolic geometry, and random media to yield rigorous existence and approximation results for tilted Brownian motion in negatively curved spaces.

Abstract

We analyze the existence of Brownian motion tilted by a potential of full support on hyperbolic spaces . 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 Schrödinger operator. On non-compact spaces like , the existence fails in general. We show that for \emph{stationary random} potentials on 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 . 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 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.
Paper Structure (10 sections, 4 theorems, 71 equations)

This paper contains 10 sections, 4 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Hyperbolic space.
  4. Brownian motion among soft traps.
  5. The periodic case.
  6. Casimir operator.
  7. The space of configurations
  8. Foliated Brownian motion.
  9. Proof of Theorem \ref{['thm:thm1']}
  10. Benjamini--Schramm limits and periodization

Key Result

Theorem 1

Let $\omega$ be a homogeneous Poisson Point Process (PPP) on ${\mathbb{H}^d}$ and let $V_\omega:{\mathbb{H}^d}\to[0,\infty)$ be a continuous factor-of-$\omega$ potential with maximum norm Then for a.e. $\omega$ and for all $x \in {\mathbb{H}^d}$, the tilted measures $\mathbf Q_T^x$ converge to a diffusion process that is independent of the starting point $x$.

Theorems & Definitions (13)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Definition 2.1
  • Example 2.2
  • Remark 2.3
  • Theorem 4: The PPP is Ramanujan
  • Remark 3.1
  • proof
  • proof : Proof of Theorem \ref{['thm:thm1']}
  • ...and 3 more