Liouville Brownian motion and quantum cones in dimension $d > 2$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866909491754172416 |
|---|---|
| author | Bertacco, Federico Gwynne, Ewain |
| author_facet | Bertacco, Federico Gwynne, Ewain |
| contents | For $d > 2$ and $γ\in (0, \sqrt{2d})$, we study the Liouville Brownian motion associated with the whole-space log-correlated Gaussian field in $\mathbb{R}^d$. We compute its spectral dimension, i.e., the short-time asymptotics of the heat kernel along the diagonal, which, in contrast to the two-dimensional case, depends on both $γ$ and on the thickness of the starting point. Furthermore, for even dimensions $d > 2$, we show that the spherical average process of the whole-space log-correlated Gaussian field in $\mathbb{R}^d$ can be identified with the integral of a stationary Gaussian Markov process of order $(d-2)/2$. Exploiting this representation, we construct the higher-dimensional analogue of the $β$-quantum cone for $β\in (-\infty, Q)$, with $Q = d/γ+ γ/2$. Lastly, for $α= Q - \sqrt{Q^2-4}$, we prove that the law of the $d$-dimensional $α$-quantum cone is invariant under shifts along the trajectories of the associated Liouville Brownian motion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_15936 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Liouville Brownian motion and quantum cones in dimension $d > 2$ Bertacco, Federico Gwynne, Ewain Probability Mathematical Physics For $d > 2$ and $γ\in (0, \sqrt{2d})$, we study the Liouville Brownian motion associated with the whole-space log-correlated Gaussian field in $\mathbb{R}^d$. We compute its spectral dimension, i.e., the short-time asymptotics of the heat kernel along the diagonal, which, in contrast to the two-dimensional case, depends on both $γ$ and on the thickness of the starting point. Furthermore, for even dimensions $d > 2$, we show that the spherical average process of the whole-space log-correlated Gaussian field in $\mathbb{R}^d$ can be identified with the integral of a stationary Gaussian Markov process of order $(d-2)/2$. Exploiting this representation, we construct the higher-dimensional analogue of the $β$-quantum cone for $β\in (-\infty, Q)$, with $Q = d/γ+ γ/2$. Lastly, for $α= Q - \sqrt{Q^2-4}$, we prove that the law of the $d$-dimensional $α$-quantum cone is invariant under shifts along the trajectories of the associated Liouville Brownian motion. |
| title | Liouville Brownian motion and quantum cones in dimension $d > 2$ |
| topic | Probability Mathematical Physics |
| url | https://arxiv.org/abs/2501.15936 |