Stochastic Flows and Marked Stable Processes

TL;DR

This work develops a continuum framework linking two coupled BESQ flows with parameter gap δ∈(0,2) to a marked, spectrally positive stable Lévy process. The two flows generate spindle regions whose widths are governed by BESQ^{−δ} excursions, and these spindles embed into the jumps of a 1+δ/2-stable Lévy process, yielding a rich space–time partition with a gasket of measure zero. The construction is mediated by a time-change and a regenerative Markov structure, enabling a Poissonian marking of spindles and a precise Lévy representation of the exploration. The theory connects to self-similar interval partition evolutions and to stable shredded disks, providing new Ray–Knight-type decompositions and bridging stochastic flows, local times, and combinatorial limit objects. Overall, the paper offers a unified approach to interlink BESQ flows, PRBM-based constructions, and continuum random structures with potential applications to random planar maps and related geometries.

Abstract

We construct a random partition of the space-time plane $\mathbb{R}_+\times \mathbb{R}$ using two coupled stochastic squared Bessel flows, whose parameters differ by $δ\in (0,2)$. We show that the cells of this partition correspond to squared Bessel excursions with a negative parameter $-δ$ which are embedded within the jumps of a spectrally positive $(1+\fracδ2)$ stable process. In particular, we demonstrate that interval partition evolutions [Forman et. al. 2020] and stable shredded disks [Björnberg, Curien and Stefánsson 2022] arise naturally in this framework.

Figures (11)

• Figure 1: Left: For a fixed point $(a_0,r_0)\in\mathbb{R}_+\times\mathbb{R}$, almost surely, a single blue/red flow line starts at $(a_0, r_0)$. Right: There exist exceptional points $(a,r)$, where the blue and red flows bifurcate and the rightmost blue line temporarily lies between the two red lines. The shaded region $\mathcal{T}_{(a,r)}$ is bounded by the rightmost blue and leftmost red lines.
• Figure 2: On the top left picture, we draw a schematic representation of the process $\xi$ colored in purple, with its jumps decorated by independent ${\rm BESQ}^{-\delta}$ excursions $f_i$ filled with different colors. On the top right picture, we draw some spindles generated by the red and blue flows $\mathcal{R}$ and $\mathcal{B}$. The widths of the spindles at level $y$, i.e. $\mathcal{B}_{r,y}(a)-\mathcal{R}_{r,y}(a-)$, are given by the values of the ${\rm BESQ}^{-\delta}$ excursions at the corresponding level, i.e. $f_i(y-\xi(t_i-))$.
• Figure 3: The left picture describes the decomposition of the white noise $\mathcal{W}$ and the flow $\mathcal{S}$ by a (forward) line $Y$ as discussed in Proposition \ref{['p:decomp']} in the case $\delta_2\le 0,\, \delta_3>0$. In this case flow lines in $\mathcal{S}^-$ will be equal to $Y$ once they hit $Y$, while flow lines in $\mathcal{S}^+$ is reflected at 0 in $[r_0,\zeta]$. The shaded area represents the noise $\mathcal{W}_Y^-$. The right picture illustrates the decomposition from a line $Y$ driven by $-\mathcal{W}^*$ as discussed in Proposition \ref{['p:dual S+']}. As shown in the proposition, flow lines in $\mathcal{S}^+$ are killed at 0 between $-\zeta^*$ and $-r_0$, while flow lines of $\mathcal{S}^-$ stays in the shaded area.
• Figure 4: In the left figure, we illustrate several regions defined in \ref{['eq:spindle']} that contain the point $(b,x)$. Among those, the spindle $\mathcal{T}_{(a,r)}$ with bottom point $(a,r)$ and top point $(c,z)$ is the maximal one. In the right figure, we depict the two dual flow lines $\mathcal{R}^*_{-x,-\cdot}(b)$ and $\mathcal{B}^*_{-x,-\cdot}(b)$ starting from $(b,x)$. These two paths satisfy $\mathcal{R}^*\le \mathcal{B}^*$ and $\mathcal{R}^*$ is absorbed at 0. By Proposition \ref{['p:ancestor equiv']}, the bottom point $(a,r)$ of the spindle $\mathcal{T}_{(a,r)}$ is the point where these two dual flow lines intersect for the last time. It is a point where both $\mathcal{R}$ and $\mathcal{B}$ bifurcate.
• Figure 5: Recall that in the definition of spindles, we include the left/red boundary, and exclude the right/blue boundary, as well as the top and bottom points. The interior of spindles is such that we also exclude the dashed red boundary in the picture. For the spindle $\mathcal{T}_{(a',r')}$, its interior is the domain filled in gray, while for the spindle $\mathcal{T}_{(a,r)}$, its interior also contains the solid red line, i.e. $\{0\}\times(y,z)$.

Theorems & Definitions (107)

• Definition 2.1: BESQ flow; aidekon2023stochastic, AWYskew
• Definition 2.2
• Proposition 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• Proposition 2.6
• Definition 2.7
• Lemma 2.8: AWYskew
• Proposition 2.9: AWYskew