The Brownian marble
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| Format: | Preprint |
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2025
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| author | Johnston, Samuel G. G. Kyprianou, Andreas Rogers, Tim Schertzer, Emmanuel |
| author_facet | Johnston, Samuel G. G. Kyprianou, Andreas Rogers, Tim Schertzer, Emmanuel |
| contents | Let $R:(0,\infty) \to [0,\infty)$ be a measurable function. Consider coalescing Brownian motions started from every point in the subset $\{ (0,x) : x \in \mathbb{R} \}$ of $[0,\infty) \times \mathbb{R}$ (with $[0,\infty)$ denoting time and $\mathbb{R}$ denoting space) and proceeding according to the following rule: the interval $\{t\} \times [L_t,U_t]$ between two consecutive Brownian motions instantaneously fragments' at rate $R(U_t - L_t)$. At a fragmentation event at a time $t$, we initiate new coalescing Brownian motions from each of the points $\{ (t,x) : x \in [L_t,U_t]\}$. The resulting process, which we call the $R$-marble, is easily constructed when $R$ is bounded, and may be considered a random subset of the Brownian web.
Under mild conditions, we show that it is possible to construct the $R$-marble when $R$ is unbounded as a limit as $n \to \infty$ of $R_n$-marbles where $R_n(g) = R(g) \wedge n$. The behaviour of this limiting process is mainly determined by the shape of $R$ near zero. The most interesting case occurs when the limit $\lim_{g \downarrow 0} g^2 R(g) = λ$ exists in $(0,\infty)$, in which case we find a phase transition. For $λ\geq 6$, the limiting object is indistinguishable from the Brownian web, whereas if $λ< 6$, then the limiting object is a nontrivial stochastic process with large gaps.
When $R(g) = λ/g^2$, the $R$-marble is a self-similar stochastic process which we refer to as the \emph{Brownian marble with parameter $λ> 0$}. We give an explicit description of the spacetime correlations of the Brownian marble, which can be described in terms of an object we call the Brownian vein; a spatial version of a recurrent extension of a killed Bessel-$3$ process. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2505_05451 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Brownian marble Johnston, Samuel G. G. Kyprianou, Andreas Rogers, Tim Schertzer, Emmanuel Probability Primary: 60K35, 82C21, 60G44, 60J65, 60J90 Let $R:(0,\infty) \to [0,\infty)$ be a measurable function. Consider coalescing Brownian motions started from every point in the subset $\{ (0,x) : x \in \mathbb{R} \}$ of $[0,\infty) \times \mathbb{R}$ (with $[0,\infty)$ denoting time and $\mathbb{R}$ denoting space) and proceeding according to the following rule: the interval $\{t\} \times [L_t,U_t]$ between two consecutive Brownian motions instantaneously fragments' at rate $R(U_t - L_t)$. At a fragmentation event at a time $t$, we initiate new coalescing Brownian motions from each of the points $\{ (t,x) : x \in [L_t,U_t]\}$. The resulting process, which we call the $R$-marble, is easily constructed when $R$ is bounded, and may be considered a random subset of the Brownian web. Under mild conditions, we show that it is possible to construct the $R$-marble when $R$ is unbounded as a limit as $n \to \infty$ of $R_n$-marbles where $R_n(g) = R(g) \wedge n$. The behaviour of this limiting process is mainly determined by the shape of $R$ near zero. The most interesting case occurs when the limit $\lim_{g \downarrow 0} g^2 R(g) = λ$ exists in $(0,\infty)$, in which case we find a phase transition. For $λ\geq 6$, the limiting object is indistinguishable from the Brownian web, whereas if $λ< 6$, then the limiting object is a nontrivial stochastic process with large gaps. When $R(g) = λ/g^2$, the $R$-marble is a self-similar stochastic process which we refer to as the \emph{Brownian marble with parameter $λ> 0$}. We give an explicit description of the spacetime correlations of the Brownian marble, which can be described in terms of an object we call the Brownian vein; a spatial version of a recurrent extension of a killed Bessel-$3$ process. |
| title | The Brownian marble |
| topic | Probability Primary: 60K35, 82C21, 60G44, 60J65, 60J90 |
| url | https://arxiv.org/abs/2505.05451 |