Weaves, webs and flows

TL;DR

This work develops a general weak convergence theory for random sets of non-crossing càdlàg paths, termed weaves, extending the Brownian web framework to arbitrary path distributions. It builds a full state-space based on a split real line and the Skorohod $M1$ topology, introduces deterministic web/flow operators, and proves that every weave decomposes into a unique pair of extremal elements (web and flow) within an equivalence class. Central contributions include a Dedekind-cut–based path-extension mechanism to connect half-infinite paths to bi-infinite flows, measurability and tightness results for random weaves, and convergence criteria expressed via finite-particle motions. The Brownian web is shown to sit inside this framework as a canonical example, with duality via a dual web capturing time-reversed structure. Overall, the paper provides a robust, distribution-free approach to universal limits of interacting-path systems and stochastic flows, with clear implications for the analysis of stochastic flows and genealogies in particle systems.

Abstract

We introduce weaves, which are random sets of non-crossing càdlàg paths that cover space-time $\overline{\mathbb{R}}\times\overline{\mathbb{R}}$. The Brownian web is one example of a weave, but a key feature of our work is that we do not assume that particle motions have any particular distribution. Rather, we present a general theory of the structure, characterization and weak convergence of weaves. We show that the space of weaves has a particularly appealing geometry, involving a partition into equivalence classes under which each equivalence class contains a pair of distinguished objects known as a web and a flow. Webs are natural generalizations of the Brownian web and the flows provide pathwise representations of stochastic flows. Moreover, there is a natural partial order on the space of weaves, characterizing the efficiency with which paths cover space-time, under which webs are precisely minimal weaves and flows are precisely maximal weaves. This structure is key to establishing weak convergence criteria for general weaves, based on weak convergence of finite collections of particle motions.

Figures (4)

• Figure 1.1.1: In both figures, time runs upwards and the spatial axis is horizontal. In each figure a weave is depicted via solid lines, with the corresponding flow depicted via including dotted lines. On the left: A weave $\mathcal{A}$, featuring a càdlàg path jumping at its initial time. Order the blue circles from bottom to top. Consider the particle motion $f_n$ starting within the $n^{\rm th}$ blue circle, which then follows the red dotted line. The limiting path $f$ is a trajectory that jumps rightwards at its initial time $t$. We require that weaves are closed sets and we require that weaves have particle motions; consequently we require that the path $f$ exists. On the right: A warning example related to Theorems \ref{['t:flow_conv']} and \ref{['t:weave_conv']}. A weave $\mathcal{A}_\epsilon$ is depicted, along with the corresponding flow $\mathcal{F}_\epsilon$ of bi-infinite paths that do not cross $\mathcal{A}_{\epsilon}$. Space-time points within the horizontal arrows, and forwards in time continuations thereof, are ramified. In the limit as $\epsilon\rightarrow 0$ the red area vanishes; the weaves $\mathcal{A}_\epsilon$ converge to a pervasive system of paths that contains crossing (jumping in both directions at $t$); the sequence of flows $(\mathcal{F}_\epsilon)_{\epsilon>0}$ are not relatively compact (due to paths that jump left-right-left between $t$ and $t+\epsilon$); whilst the $m$-particle motions $\mathcal{A}_\epsilon|_{\boldsymbol{z}}=\mathcal{F}_\epsilon|_{\boldsymbol{z}}$ from finite sets $\boldsymbol{z}$ of non-ramified points converge to those of a weave (which contains only the leftwards jump at $t$).
• Figure 2.2.1: On the left, the closed graph $G(f)$ and interpolated graph $H(f)$ of a path $f$. On the right, the compactification $\mathbb{R}^2_{\rm c}$ of $\overline\mathbb{R}\times\mathbb{R}$ with the interpolated graph of $f$, with some points marked for convenience. In FreemanSwart2023$\mathbb{R}^2_c$ is referred to as a squeezed space.
• Figure 2.5.1: In all three figures, time runs upwards and the spatial axis is horizontal. In each figure a weave is depicted via solid lines, with the corresponding flow depicted via including dotted lines. On the left: A weave $\mathcal{A}$ such that $\mathcal{A}(D)$ contains the bi-infinite red path, whereas $\mathcal{A}|_D$ does not. Here $D\subseteq\mathbb{R}^2$ must be dense and non-ramified with respect to the weave $\mathcal{A}$. In this case we have that $\mathop{\mathrm{web}}\nolimits(\mathcal{A})=\overline{(\mathcal{A}|_D)_\uparrow}\prec\overline{\mathcal{A}(D)_\uparrow}$. On the center and right: An example related to continuity of the $\mathop{\mathrm{flow}}\nolimits(\cdot)$ map and lack of continuity of the $\mathop{\mathrm{web}}\nolimits(\cdot)$ map, as well as to the existence of isolated points within flows and general weaves. In the center, the weave $\mathcal{A}_\epsilon$ and corresponding flow $\mathcal{F}_\epsilon=\mathop{\mathrm{flow}}\nolimits(\mathcal{A}_\epsilon)$ are depicted. The limiting weave $\mathcal{A}=\lim_{\epsilon\rightarrow 0}\mathcal{A}_\epsilon$ is depicted on the right, with $\mathcal{F}=\mathop{\mathrm{flow}}\nolimits(\mathcal{A})$ again depicted via including dotted lines. The red paths collapse to a single bi-infinite path in the limit. Note that $\mathcal{F}_\epsilon\rightarrow\mathcal{F}$ in accordance with part 3 of Theorem \ref{['t:flow_conv']}. In this case $\mathcal{W}_\epsilon=\mathop{\mathrm{web}}\nolimits(\mathcal{A}_\epsilon)$ is equal to $(\mathcal{A}_\epsilon)_\uparrow$. Therefore $\mathcal{W}_\epsilon\rightarrow\mathcal{A}_\uparrow$. On the right, note that $\mathcal{W}=\mathop{\mathrm{web}}\nolimits(\mathcal{A})$ does not include the bi-infinite red path, and that every point of this path ramified. Consequently $\lim_{\epsilon\rightarrow 0}\mathcal{W}_\epsilon\neq \mathcal{W}$, showing that the map $\mathop{\mathrm{web}}\nolimits(\cdot)$ is discontinuous at $\mathcal{W}$. Note that the bi-infinite red path on the right is an isolated point of $\mathcal{A}$ but is not an isolated point of $\mathcal{F}$.
• Figure 3.1.1: On the left: A schematic depiction of the subsets $L(f)$ and $R(f)$ of $\overline\mathbb{R}\times\mathbb{R}_\mathfrak{s}$. Time is running upwards, with values taken by $f(t\star)$ shown as a thick black line. To help visualize, when $f$ jumps at time $t$ we depict time as split into $t-$ and $t+$ via thin horizontal lines. The path $f$ makes a jump at its starting time $\sigma_f$, at the very bottom of the figure. Note that $L_{\sigma_f-}(f)=\emptyset$ and $R_{\sigma_f-}(f)=(f(\sigma_f-),\infty]$ because the jump at $\sigma_f$ is rightwards. On the right: Four examples of sets $\{f,g\}$ containing two paths. In each example, the initial point of $g$ lies to the left of the initial point of $f$. The paths in (i) satisfy $f\vartriangleleft g$ and do not cross, but in examples (ii)--(iv) we have that $f$ crosses $g$ from right to left.

Theorems & Definitions (93)

• Lemma 2.1.1
• Definition 2.1.2
• Proposition 2.2.1
• Definition 2.3.1
• Definition 2.4.1
• Definition 2.4.2
• Theorem 2.4.3
• Theorem 2.4.4
• Theorem 2.4.5
• Theorem 2.4.6