Brownian couplings, convexity, and shy-ness

Benjamini, Burdzy, and Chen (2007) introduced the notion of a shy coupling: a coupling of a Markov process such that, for suitable starting points, there is a positive chance of the two component processes of the coupling staying at least a given positive distance away from each other for all time. Among other results, they showed that no shy couplings could exist for reflected Brownian motions in C-2 bounded convex planar domains whose boundaries contain no line segments. Here we use potential-theoretic methods to extend this Benjamini et al. (2007) result (a) to all bounded convex domains (whether planar and smooth or not) whose boundaries contain no line segments, (b) to all bounded convex planar domains regardless of further conditions on the boundary

Return to the Poissonian City

Consider the following random spatial network: in a large disk, construct a network using a stationary and isotropic Poisson line process of unit intensity. Connect pairs of points using the network, with initial / final segments of the connecting path formed by travelling off the network in the opposite direction to that of the destination / source. Suppose further that connections are established using "near-geodesics", constructed between pairs of points using the perimeter of the cell containing these two points and formed using only the Poisson lines not separating them. If each pair of points generates an infinitesimal amount of traffic divided equally between the two connecting near-geodesics, and if the Poisson line pattern is conditioned to contain a line through the centre, then what can be said about the total flow through the centre? In earlier work ("Geodesics and flows in a Poissonian city", Annals of Applied Probability, 21(3), 801--842, 2011) it was shown that a scaled version of this flow had asymptotic distribution given by the 4-volume of a region in 4-space, constructed using an improper anisotropic Poisson line process in an infinite planar strip. Here we construct a more amenable representation in terms of two "seminal curves" defined by the improper Poisson line process, and establish results which produce a framework for effective simulation from this distribution up to an L1 error which tends to zero with increasing computational effort.Comment: 11 pages, 2 figures Various minor edits, corrections to multiplicative constants in Theorem 5.1. Version 2: minor stylistic corrections, added acknowledgement of grant support. Version 3: three further minor corrections. This paper is due to appear in Journal of Applied Probability, Volume 51

Coupling, local times, immersions

This paper answers a question of \'{E}mery [In S\'{e}minaire de Probabilit\'{e}s XLII (2009) 383-396 Springer] by constructing an explicit coupling of two copies of the Bene\v{s} et al. [In Applied Stochastic Analysis (1991) 121-156 Gordon & Breach] diffusion (BKR diffusion), neither of which starts at the origin, and whose natural filtrations agree. The paper commences by surveying probabilistic coupling, introducing the formal definition of an immersed coupling (the natural filtration of each component is immersed in a common underlying filtration; such couplings have been described as co-adapted or Markovian in older terminologies) and of an equi-filtration coupling (the natural filtration of each component is immersed in the filtration of the other; consequently the underlying filtration is simultaneously the natural filtration for each of the two coupled processes). This survey is followed by a detailed case-study of the simpler but potentially thematic problem of coupling Brownian motion together with its local time at $0$. This problem possesses its own intrinsic interest as well as being closely related to the BKR coupling construction. Attention focusses on a simple immersed (co-adapted) coupling, namely the reflection/synchronized coupling. It is shown that this coupling is optimal amongst all immersed couplings of Brownian motion together with its local time at $0$, in the sense of maximizing the coupling probability at all possible times, at least when not started at pairs of initial points lying in a certain singular set. However numerical evidence indicates that the coupling is not a maximal coupling, and is a simple but non-trivial instance for which this distinction occurs. It is shown how the reflection/synchronized coupling can be converted into a successful equi-filtration coupling, by modifying the coupling using a deterministic time-delay and then by concatenating an infinite sequence of such modified couplings. The construction of an explicit equi-filtration coupling of two copies of the BKR diffusion follows by a direct generalization, although the proof of success for the BKR coupling requires somewhat more analysis than in the local time case.Comment: Published at http://dx.doi.org/10.3150/14-BEJ596 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

From Random Lines to Metric Spaces

Consider an improper Poisson line process, marked by positive speeds so as to satisfy a scale-invariance property (actually, scale-equivariance). The line process can be characterized by its intensity measure, which belongs to a one-parameter family if scale and Euclidean invariance are required. This paper investigates a proposal by Aldous, namely that the line process could be used to produce a scale-invariant random spatial network (SIRSN) by means of connecting up points using paths which follow segments from the line process at the stipulated speeds. It is shown that this does indeed produce a scale-invariant network, under suitable conditions on the parameter; indeed that this produces a parameter-dependent random geodesic metric for d-dimensional space ($d\geq2$), where geodesics are given by minimum-time paths. Moreover in the planar case it is shown that the resulting geodesic metric space has an almost-everywhere-unique-geodesic property, that geodesics are locally of finite mean length, and that if an independent Poisson point process is connected up by such geodesics then the resulting network places finite length in each compact region. It is an open question whether the result is a SIRSN (in Aldous' sense; so placing finite mean length in each compact region), but it may be called a pre-SIRSN.Comment: Version 1: 46 pages, 10 figures Version 2: 47 pages, 10 figures (various typos and stylistic amendments, added dedication to Burkholder, added references concerning Lipschitz property and Sobolev space

Limit theorems for empirical Fréchet means of independent and non-identically distributed manifold-valued random variables

We prove weak laws of large numbers and central limit theorems of Lindeberg type for empirical centres of mass (empirical Fréchet means) of independent nonidentically distributed random variables taking values in Riemannian manifolds. In order to prove these theorems we describe and prove a simple kind of Lindeberg–Feller central approximation theorem for vector-valued random variables, which may be of independent interest and is therefore the subject of a self-contained section. This vector-valued result allows us to clarify the number of conditions required for the central limit theorem for empirical Fréchet means, while extending its scope

Coupling of Brownian motions in Banach spaces

Consider a separable Banach space $\mathcal{W}$ supporting a non-trivial Gaussian measure $\mu$. The following is an immediate consequence of the theory of Gaussian measure on Banach spaces: there exist (almost surely) successful couplings of two $\mathcal{W}$-valued Brownian motions $\mathbf{B}$ and $\widetilde{\mathbf{B}}$ begun at starting points $\mathbf{B}(0)$ and $\widetilde{\mathbf{B}}(0)$ if and only if the difference $\mathbf{B}(0)-\widetilde{\mathbf{B}}(0)$ of their initial positions belongs to the Cameron-Martin space $\mathcal{H}_{\mu}$ of $\mathcal{W}$ corresponding to $\mu$. For more general starting points, can there be a "coupling at time $\infty$", such that almost surely $\|\mathbf{B}(t)-\widetilde{\mathbf{B}}(t)\|_{\mathcal{W}} \to 0$ as $t\to\infty$? Such couplings exist if there exists a Schauder basis of $\mathcal{W}$ which is also a $\mathcal{H}_{\mu}$-orthonormal basis of $\mathcal{H}_{\mu}$. We propose (and discuss some partial answers to) the question, to what extent can one express the probabilistic Banach space property "Brownian coupling at time $\infty$ is always possible" purely in terms of Banach space geometry?Comment: 12 page

Coupling iterated Kolmogorov diffusions

The Kolmogorov (1934) diffusion is the two-dimensional diffusion generated by real Brownian motion B and its time integral integral B d t. In this paper we construct successful co-adapted couplings for iterated Kolmogorov diffusions defined by adding iterated time integrals integral integral B d s d t,... as further components to the original Kolmogorov diffusion. A Laplace-transform argument shows it is not possible successfully to couple all iterated time integrals at once; however we give an explicit construction of a successful co-adapted coupling method for (B, integral B d t, integral integral B d s d t); and a more implicit construction of a successful co-adapted coupling method which works for finite sets of iterated time integrals

Perfect Simulation of M/G/cM/G/c Queues

In this paper we describe a perfect simulation algorithm for the stable $M/G/c$ queue. Sigman (2011: Exact Simulation of the Stationary Distribution of the FIFO M/G/c Queue. Journal of Applied Probability, 48A, 209--213) showed how to build a dominated CFTP algorithm for perfect simulation of the super-stable $M/G/c$ queue operating under First Come First Served discipline, with dominating process provided by the corresponding $M/G/1$ queue (using Wolff's sample path monotonicity, which applies when service durations are coupled in order of initiation of service), and exploiting the fact that the workload process for the $M/G/1$ queue remains the same under different queueing disciplines, in particular under the Processor Sharing discipline, for which a dynamic reversibility property holds. We generalize Sigman's construction to the stable case by comparing the $M/G/c$ queue to a copy run under Random Assignment. This allows us to produce a naive perfect simulation algorithm based on running the dominating process back to the time it first empties. We also construct a more efficient algorithm that uses sandwiching by lower and upper processes constructed as coupled $M/G/c$ queues started respectively from the empty state and the state of the $M/G/c$ queue under Random Assignment. A careful analysis shows that appropriate ordering relationships can still be maintained, so long as service durations continue to be coupled in order of initiation of service. We summarize statistical checks of simulation output, and demonstrate that the mean run-time is finite so long as the second moment of the service duration distribution is finite.Comment: 28 pages, 5 figure