Weak convergence of the localized disturbance flow to the coalescing Brownian flow

Norris, James R.; Turner, Amanda

doi:10.1214/13-aop845

Cited by 18 publications

(64 citation statements)

References 17 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…An outstanding property of Konarovskyi's process is the fact that, for a large family of initial measures, it takes values in the set of measures with finite support for each time t > 0 (see [11]), whereas the values of the Wasserstein diffusion of von Renesse and Sturm are probability measures on [0, 1] with no absolutely continuous part and no discrete part. The model introduced by Konarovskyi is a modification of the Arratia flow, also called Coalescing Brownian flow, introduced by Arratia [3] and subject of many interest, among others in [7], [14], [16], [17]. It consists of Brownian particles starting at discrete points of the real line and moving independently until they meet another particle: when they meet, they stick together to form a single Brownian particle.…”

Section: Konarovskyi's Modelmentioning

confidence: 99%

A new approach for the construction of a Wasserstein diffusion

Marx¹

2018

Electron. J. Probab.

View full text Add to dashboard Cite

We propose in this paper a construction of a diffusion process on the space P 2 (R) of probability measures with a second-order moment. This process was introduced in several papers by Konarovskyi (see e.g. [12]) and consists of the limit as N tends to +∞ of a system of N coalescing and mass-carrying particles. It has properties analogous to those of a standard Euclidean Brownian motion, in a sense that we will precise in this paper. We also compare it to the Wasserstein diffusion on P 2 (R) constructed by von Renesse and Sturm in [22]. We obtain that process by the construction of a system of particles having shortrange interactions and by letting the range of interactions tend to zero. This construction can be seen as an approximation of the singular process of Konarovskyi by a sequence of smoother processes. conducted by Ambrosio, Gigli, Savare, Villani, Lions and many others ([1], [5], [15], [20], [21]), which led to important improvements in optimal transport theory. Second, the transition costs of the Wasserstein diffusion are given by a Varadhan formula (see [22], Corollary 7.19). The formula is identical to the Euclidean case, up to the replacement of the Euclidean norm by d W .Although the existence of a Wasserstein diffusion was initially proven by von Renesse and Sturm using Dirichlet processes and the theory of Dirichlet forms (see [8]), it can also be obtained as a limit of finite-dimensional systems of interacting particles, see [2] and [19]. Nevertheless, we will focus in this paper on a construction of a system of particles which seems more natural and simpler and which is due to Konarovskyi in [10] and [12]. (iv) for all u, u ∈ [0, 1], y(u, ·), y(u , ·) t = t 0 1 {τ u,u s} m(u, s) ds, where m(u, t) = 1 0 1 {∃s t: y(u,s)=y(v,s)} dv and τ u,u = inf{t 0 : y(u, t) = y(u , t)} ∧ T .By transporting the Lebesgue measure on [0, 1] by the map y(·, t), we obtain a measurevalued process (µ t ) t∈[0,T ] defined by: µ t := Leb | [0,1] • y(·, t) −1 . In other words, u → y(u, t) is the quantile function associated to µ t . An important feature of this process is that for each positive t, µ t is an atomic measure with a finite number of atoms, or in other words that y(·, t) is a step function.More generally, Konarovskyi proves in [11] that this construction also holds for a greater family of initial measures µ 0 . He constructs a process y g in D([0, 1], C[0, T ]) satisfying (ii) − (iv) and:(i) for all u ∈ [0, 1], y g (u, 0) = g(u), for every non-decreasing càdlàg function g from [0, 1] into R such that there exists p > 2 satisfying 1 0 |g(u)| p du < ∞. In other words, he generalizes the construction of a diffusion starting from any probability measure µ 0 satisfying R |x| p dµ 0 (x) < ∞ for a certain p > 2, where µ 0 = Leb | [0,1] • g −1 , which means that g is the quantile function of the initial measure. The property that y g (·, t) is a step function for each t > 0 remains true for this larger class of functions g.The process y g is said to be coalescent: almost surely, for every u, v ∈ [0, 1] and for every ...

show abstract

Section: Konarovskyi's Modelmentioning

confidence: 99%

A new approach for the construction of a Wasserstein diffusion

Marx¹

2018

Electron. J. Probab.

View full text Add to dashboard Cite

show abstract

“…It was shown in [NT15] that C Compared with the paths topology introduced in Section 2.1, the advantage of the weak flow topology is that it is more natural for studying stochastic flows, and it allows for discontinuity in the flow lines (paths). The limitation is that it is restricted to flows with non-crossing paths.…”

Section: Weak Flow Topologymentioning

confidence: 99%

“…Interestingly, the image of ∂K 0 under F n forms a coalescing flow on the circle, where for x, y ∈ ∂K 0 ⊂ ∂K n , the length of the arc between F n (x) and F n (y) on the unit circle is proportional to the probability that a new particle will be attached to the corresponding part of ∂K n between x and y. In the limit that the particle radius δ ↓ 0, while time is sped up by a factor of δ −3 , this coalescing flow can be seen as a localized disturbance flow studied in [NT15], which converges to the coalescing Brownian flow on the circle Figure 19: From left to right, illustration of paths in the drainage network of [GRS04], the drainage network of [RSS16a], Poisson trees [FLT04], the directed spanning forest [BB07], and the radial spanning tree [BB07]. w.r.t.…”

Section: Planar Aggregation Modelsmentioning

confidence: 99%

The Brownian Web, the Brownian Net, and their Universality

Schertzer¹,

Sun

Swart

2016

Advances in Disordered Systems, Random Processes and Some Applications

107

View full text Add to dashboard Cite

The Brownian web is a collection of one-dimensional coalescing Brownian motions starting from everywhere in space and time, and the Brownian net is a generalization that also allows branching. They appear in the diffusive scaling limits of many one-dimensional interacting particle systems with branching and coalescence. This article gives an introduction to the Brownian web and net, and how they arise in the scaling limits of various one-dimensional models, focusing mainly on coalescing random walks and random walks in i.i.d. space-time random environments. We will also briefly survey models and results connected to the Brownian web and net, including alternative topologies, population genetic models, true self-repelling motion, planar aggregation, drainage networks, oriented percolation, black noise and critical percolation. Some open questions are discussed at the end.MSC 2000. Primary: 82C21 ; Secondary: 60K35, 60D05.

show abstract

“…In 1998 the physicists M. Hastings and L. Levitov introduced a one-parameter family of continuum models for growing clusters (K n ) n≥0 on the plane [13], which can be considered as an off-lattice version of discrete planar aggregation models such as the Eden model or Diffusion Limited Aggregation (DLA). In this paper we focus on the simplest of these models, so called HL(0), which has proven to be already very rich from a mathematical point of view, and has received much attention in recent years [27,26,25,18].…”

Section: Introductionmentioning

confidence: 99%

Fluctuation results for Hastings–Levitov planar growth

Silvestri

2015

Probab. Theory Relat. Fields

View full text Add to dashboard Cite

We study the fluctuations of the outer domain of Hastings-Levitov clusters in the small particle limit. These are shown to be given by a continuous Gaussian process F taking values in the space of holomorphic functions on {|z| > 1}, of which we provide an explicit construction.The boundary values W of F are shown to perform an Ornstein-Uhlenbeck process on the space of distributions on the unit circle T, which can be described as the solution to the stochastic fractional heat equationwhere ∆ denotes the Laplace operator acting on the spatial component, and ξ(t, ϑ) is a spacetime white noise. As a consequence we find that, when the cluster is left to grow indefinitely, the boundary process W converges to a log-correlated Fractional Gaussian Field, which can be realised as (−∆) −1/4 W , for W complex White Noise on T.

show abstract

Weak convergence of the localized disturbance flow to the coalescing Brownian flow

Cited by 18 publications

References 17 publications

A new approach for the construction of a Wasserstein diffusion

A new approach for the construction of a Wasserstein diffusion

The Brownian Web, the Brownian Net, and their Universality

Fluctuation results for Hastings–Levitov planar growth

Contact Info

Product

Resources

About