Extending the noise of splitting to its completion and stability of Brownian maxima
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866918223048343552 |
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| author | Vidmar, Matija Warren, Jon |
| author_facet | Vidmar, Matija Warren, Jon |
| contents | The stochastic noise of splitting, defined initially on the (basic) algebra of finite unions of intervals of the real line, is extended to a largest class of domains. The $σ$-fields of this largest extension constitute the completion, in the sense of noise-type Boolean algebras, of the range of the unextended (basic) noise. The basic noise extends to a given measurable domain precisely when a certain stability property is met: the times at which a Brownian motion has local maxima which fall inside the domain must remain unaffected under resampling of the Brownian increments outside the domain; together with the same being true for the complement of the domain. A set that is equal to an open set modulo a Lebesgue negligible one, with the same holding of its complement, has this stability property, but others have it too: the extension is non-trivial. Some domains are totally unstable with respect to the indicated resampling, and to them the extension cannot be made. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_05144 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Extending the noise of splitting to its completion and stability of Brownian maxima Vidmar, Matija Warren, Jon Probability primary: 60G20, secondary: 60J65, 60G05 The stochastic noise of splitting, defined initially on the (basic) algebra of finite unions of intervals of the real line, is extended to a largest class of domains. The $σ$-fields of this largest extension constitute the completion, in the sense of noise-type Boolean algebras, of the range of the unextended (basic) noise. The basic noise extends to a given measurable domain precisely when a certain stability property is met: the times at which a Brownian motion has local maxima which fall inside the domain must remain unaffected under resampling of the Brownian increments outside the domain; together with the same being true for the complement of the domain. A set that is equal to an open set modulo a Lebesgue negligible one, with the same holding of its complement, has this stability property, but others have it too: the extension is non-trivial. Some domains are totally unstable with respect to the indicated resampling, and to them the extension cannot be made. |
| title | Extending the noise of splitting to its completion and stability of Brownian maxima |
| topic | Probability primary: 60G20, secondary: 60J65, 60G05 |
| url | https://arxiv.org/abs/2407.05144 |