Quenched path limits and periodization stability for tilted Brownian motion in Poissonian potentials on $\mathbb{H}^d$
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| Format: | Preprint |
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2026
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| author | Abert, Miklos Arras, Adam Kim, Jaelin |
| author_facet | Abert, Miklos Arras, Adam Kim, Jaelin |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_18433 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Quenched path limits and periodization stability for tilted Brownian motion in Poissonian potentials on $\mathbb{H}^d$ Abert, Miklos Arras, Adam Kim, Jaelin Probability Differential Geometry Spectral Theory Primary 60J60, Secondary 60G55, 58J65, 35P15, 53C35, 60J50 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. |
| title | Quenched path limits and periodization stability for tilted Brownian motion in Poissonian potentials on $\mathbb{H}^d$ |
| topic | Probability Differential Geometry Spectral Theory Primary 60J60, Secondary 60G55, 58J65, 35P15, 53C35, 60J50 |
| url | https://arxiv.org/abs/2602.18433 |