Weaves, webs and flows (Q6595717)

The authors introduce a rich and natural class of objects that generalize the Brownian web (these objects are called weaves). Informally, a weave is a random set of non-crossing càdlàg paths, such that each point of space-time is almost surely touched by at least one path. The paths take values in \(\overline{\mathbb{R}}\) and each path runs until time \(+\infty\), but paths may begin at any point of space-time. It is most important to note what is missing: the paths follow any particular distribution. Weaves consisting exclusively of bi-infinite paths provide natural path-wise representations of (sufficiently regular) stochastic flows. It is remarkable that, despite their long history, stochastic flows have struggled to give rise to a theory of their own weak convergence. The underlying problem is that stochastic flows have traditionally been given a ``point-wise'' representation, where for each pair of times \(-\infty < s < t < \infty\) a random function \(X_{s,t}: \mathbb{R} \to\mathbb{R}\) represents the movement of particles during \([s; t]\). Thus \(X_{s,t}(x)\) denotes the position at time \(t\) of the particle that, at time \(s\), was at location \(x\). \N\NThe modern perspective is to view a stochastic process as a single random variable, whose value is a random path. Such a representation is known as a ``path-wise'' representation. The same principle will apply to random sets of càdlàg paths, however such objects have not yet made an analogous transition -- with the exception of the Brownian web and its close relatives. \N\NThe present paper seeks to remedy this situation. Loosely, one such particle begins at each point of space-time, and each particle gives rise to a half-infinite path within the web. \textit{L. R. G. Fontes} et al. [Ann. Probab. 32, No. 4, 2857--2883 (2004; Zbl 1105.60075)] represented the Brownian web as a (single) random variable whose value is a random set of continuous paths, and introduced the term Brownian web. In this representation they gave the first conditions for weak convergence to the Brownian web, based on the forwards-in-time motions of finite sets of particles. A large body of literature has since emerged, leading to the refined criteria available in the survey of \textit{E. Schertzer} et al. [``The Brownian web, the Brownian net, and their universality'', Preprint, \url{arXiv:1506.00724}]). Close relatives of the Brownian web have been investigated in similar style and the Brownian web is understood to be the scaling limit of a large and diverse universality class. The key to this success has been the availability of good criteria for characterization and weak convergence. Such criteria must strike a careful balance: a type of convergence that preserves less information is often easier to prove, and is more often true, but is also less meaningful. One possible strategy is to map sets of paths to other objects in order to induce a topology that may be used as a basis for weak convergence. This strategy was used by \textit{N. Berestycki} et al. [Ann. Probab. 43, No. 6, 3177--3215 (2015; Zbl 1345.60111)] and \textit{G. Cannizzaro} and \textit{M. Hairer} [Electron. J. Probab. 28, Paper No. 102, 47 p. (2023; Zbl 07733587)] for the case of continuous coalescing paths, respectively mapping to sets of ``tubes'' and real trees (see Section 2.7 for more details). \N\NLet us now briefly comment on the significance of webs. An exploration of the space of weaves will uncover a natural partition into equivalence classes. Each equivalence class features two distinguished elements, one of which is a flow (as discussed above) and the other of which as a web. The authors see that the property of being a web is equivalent to what remains if one takes the usual definition of the Brownian web and removes the requirement that the particle motions have a particular distribution. Webs and flows are in bijective correspondence; moreover they are the extremal points, respectively minima and maxima, within a structure that we will shortly describe. Within much of the literature on the Brownian web, the proofs rely heavily on the distribution of coalescing Brownian motions. Consequently, arguments have little in common. Despite this, the authors remark that what is known about the Brownian web has been invaluable in writing the present article, and the Brownian web is a canonical example of a weave. In fact most of the results are new even in the special case of the Brownian web. Section 2 introduces a state space and, following that, give rigorous statements of the main results. A central concept is the \(m\)-particle motion of a weave. It is clear a priori that a pair of càdlàg paths might cross each other. Moreover a weave is a flow if and only if it comprises exclusively of bi-infinite paths. Therefore, flows provide natural path-wise representations of stochastic flows. This structure is key to establishing weak convergence criteria for general weaves, based on weak convergence of finite collections of particle motions.