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| Main Author: | |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2305.00734 |
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| _version_ | 1866917791151423488 |
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| author | Ooi, Takumu |
| author_facet | Ooi, Takumu |
| contents | As represented by the Liouville measure, Gaussian multiplicative chaos is a random measure constructed from a Gaussian field. Under certain technical assumptions, we prove the convergence of a process time-changed by Gaussian multiplicative chaos in the case the latter object is square integrable (the $L^2$-regime). As examples of the main result, we prove that, in the whole $L^2$-regime, the scaling limit of the Liouville simple random walk on $\mathbb{Z}^2$ is Liouville Brownian motion and, as $α\to 1$, Liouville $α$-stable processes on $\mathbb{R}$ converge weakly to the Liouville Cauchy process. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2305_00734 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Convergence of processes time-changed by Gaussian multiplicative chaos Ooi, Takumu Probability 60K37, 31C25, 60J55, 60G57, 60G60 As represented by the Liouville measure, Gaussian multiplicative chaos is a random measure constructed from a Gaussian field. Under certain technical assumptions, we prove the convergence of a process time-changed by Gaussian multiplicative chaos in the case the latter object is square integrable (the $L^2$-regime). As examples of the main result, we prove that, in the whole $L^2$-regime, the scaling limit of the Liouville simple random walk on $\mathbb{Z}^2$ is Liouville Brownian motion and, as $α\to 1$, Liouville $α$-stable processes on $\mathbb{R}$ converge weakly to the Liouville Cauchy process. |
| title | Convergence of processes time-changed by Gaussian multiplicative chaos |
| topic | Probability 60K37, 31C25, 60J55, 60G57, 60G60 |
| url | https://arxiv.org/abs/2305.00734 |