Markovian Bridges: Construction, Palm Interpretation, and Splicing
01 Jan 1993-pp 101-134
TL;DR: In this paper, the authors define a Markovian bridge as a process obtained by conditioning the Markov process X to start in some state x at time 0 and arrive at some state z at time t. Once the definition is made precise, they call this process the (x, t, z)-bridge derived from X.
Abstract: By a Markovian bridge we mean a process obtained by conditioning a Markov process X to start in some state x at time 0 and arrive at some state z at time t. Once the definition is made precise, we call this process the (x, t, z)-bridge derived from X. Important examples are provided by Brownian and Bessel bridges, which have been extensively studied and find numerous applications. See for example [PY1,SW,Sa,H,EL,AP,BP].It is part of Markovian folklore that the right way to define bridges in some generality is by a suitable Doob h -transform of the space-time process. This method was used by Getoor and Sharpe [GS4] for excursion bridges, and by Salminen [Sa] for one-dimensional diffusions, but the idea of using h-transforms to construct bridges seems to be much older. Our first object in this paper is to make this definition of bridges precise in a suitable degree of generality, with the aim of dispelling all doubts about the existence of clearly defined bridges for nice Markov processes. This we undertake in Section 2. In Section 3 we establish a conditioning formula involving bridges and continuous additive functionals of the Markov process. This formula can be found in [RY, Ex. (1.16) of Ch. X, p.378] under rather stringent continuity conditions. One of our goals here is to prove the formula in its “natural” setting. We apply the conditioning formula in Section 4 to show how Markovian bridges are involved in a family of Palm distributions associated with continuous additive functionals of the Markov process. This generalizes an approach to bridges suggested in a particular case by Kallenberg [K1], and connects this approach to the more conventional definition of bridges adopted here.
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01 Jan 2006
TL;DR: In this paper, the Brownian forest and the additive coalescent were constructed for random walks and random forests, respectively, and the Bessel process was used for random mappings.
Abstract: Preliminaries.- Bell polynomials, composite structures and Gibbs partitions.- Exchangeable random partitions.- Sequential constructions of random partitions.- Poisson constructions of random partitions.- Coagulation and fragmentation processes.- Random walks and random forests.- The Brownian forest.- Brownian local times, branching and Bessel processes.- Brownian bridge asymptotics for random mappings.- Random forests and the additive coalescent.
1,371 citations
Cites background or methods from "Markovian Bridges: Construction, Pa..."
...These definitions of conditioned Brownian motions have been made rigorous in a number of ways: for instance by the method of Doob h-transforms [255, 394, 155], and as weak limits as ε ↓ 0 of the distribution of B given suitable events Aε, as in [124, 69], for instance...
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...The following lemma is a well known consequence of Itô’s description of excursions of a Markov process and the general construction of bridges of a nice recurrent Markov process, as in [155]....
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TL;DR: In this paper, it was shown that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n−1/4, converge as n → ∞ in distribution for the Gromov-Hausdorff topology to a limiting metric space.
Abstract: We prove that uniform random quadrangulations of the sphere with n faces, endowed with the usual graph distance and renormalized by n−1/4, converge as n → ∞ in distribution for the Gromov–Hausdorff topology to a limiting metric space. We validate a conjecture by Le Gall, by showing that the limit is (up to a scale constant) the so-called Brownian map, which was introduced by Marckert–Mokkadem and Le Gall as the most natural candidate for the scaling limit of many models of random plane maps. The proof relies strongly on the concept of geodesic stars in the map, which are configurations made of several geodesics that only share a common endpoint and do not meet elsewhere.
374 citations
TL;DR: In this article, the authors studied the tree-valued Markov process (G u, 0 ≤ u ≤ 1) and an analogous process in which G 1 ∗ is a critical or subcritical Galton-Watson tree conditioned to be infinite.
Abstract: Let G be a Galton-Watson tree, and for 0 ≤ u ≤ 1 let G u be the subtree of G obtained by retaining each edge with probability u. We study the tree-valued Markov process ( G u, 0 ≤ u ≤ 1) and an analogous process ( G u∗, 0 ≤ u ≤ 1) in which G 1∗ is a critical or subcritical Galton-Watson tree conditioned to be infinite. Results simplify and are further developed in the special case of Poisson offspring distribution.
147 citations
TL;DR: In this paper, a self-similar Markov process X is constructed via its associated entrance law, which can be viewed as X conditioned never to hit 0, and then the process is constructed similarly to the way in which the Brownian excursion measure is constructed through the law of a Bessel(3) process.
Abstract: Let ξ be a real-valued Levy process that satisfies Cramer's condition, and X a self-similar Markov process associated with ξ via Lamperti's transformation. In this case, X has 0 as a trap and satisfies the assumptions set out by Vuolle-Apiala. We deduce from the latter that there exists a unique excursion measure \exc, compatible with the semigroup of X and such that \exc(X0+>0)=0. Here, we give a precise description of \exc via its associated entrance law. To this end, we construct a self-similar process X
atural, which can be viewed as X conditioned never to hit 0, and then we construct \exc similarly to the way in which the Brownian excursion measure is constructed via the law of a Bessel(3) process. An alternative description of \exc is given by specifying the law of the excursion process conditioned to have a given length. We establish some duality relations from which we determine the image under time reversal of \exc.
129 citations
Posted Content•
TL;DR: In this article, a partiton-based Fubini calculus for Poisson processes is discussed, which is an amplification of Bayesian techniques developed in Lo and Weng for gamma/Dirichlet processes, and an explicit partition based calculus is then developed for such models, which also includes a series of important exponential change of measure formula.
Abstract: This article discusses the usage of a partiton based Fubini calculus for Poisson processes. The approach is an amplification of Bayesian techniques developed in Lo and Weng for gamma/Dirichlet processes. Applications to models are considered which all fall within an inhomogeneous spatial extension of the size biased framework used in Perman, Pitman and Yor. Among some of the results; an explicit partition based calculus is then developed for such models, which also includes a series of important exponential change of measure formula. These results are applied to obtain results for Levy-Cox models, identities related to the two-parameter Poisson-Dirichlet process and other processes, generalisations of the Markov-Krein correspondence, calculus for extended Neutral to the Right processes, among other things.
121 citations
References
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Book•
01 Jan 1990
TL;DR: In this article, the authors present a comprehensive survey of the literature on limit theorems in distribution in function spaces, including Girsanov's Theorem, Bessel Processes, and Ray-Knight Theorem.
Abstract: 0. Preliminaries.- I. Introduction.- II. Martingales.- III. Markov Processes.- IV. Stochastic Integration.- V. Representation of Martingales.- VI. Local Times.- VII. Generators and Time Reversal.- VIII. Girsanov's Theorem and First Applications.- IX. Stochastic Differential Equations.- X. Additive Functionals of Brownian Motion.- XI. Bessel Processes and Ray-Knight Theorems.- XII. Excursions.- XIII. Limit Theorems in Distribution.- 1. Gronwall's Lemma.- 2. Distributions.- 3. Convex Functions.- 4. Hausdorff Measures and Dimension.- 5. Ergodic Theory.- 6. Probabilities on Function Spaces.- 7. Bessel Functions.- 8. Sturm-Liouville Equation.- Index of Notation.- Index of Terms.- Catalogue.
7,338 citations
Book•
01 Apr 1986
TL;DR: In this paper, a broad cross-section of the literature available on one-dimensional empirical processes is summarized, with emphasis on real random variable processes as well as a wide-ranging selection of applications in statistics.
Abstract: Here is the first book to summarize a broad cross-section of the large volume of literature available on one-dimensional empirical processes. Presented is a thorough treatment of the theory of empirical processes, with emphasis on real random variable processes as well as a wide-ranging selection of applications in statistics. Featuring many tables and illustrations accompanying the proofs of major results, coverage includes foundations - special spaces and special processes, convergence and distribution of empirical processes, alternatives and processes of residuals, integral tests of fit and estimated empirical processes and martingale methods.
2,774 citations
Book•
01 Jan 1967
TL;DR: The Borel subsets of a metric space Probability measures in the metric space and probability measures in a metric group Probability measure in locally compact abelian groups The Kolmogorov consistency theorem and conditional probability probabilistic probability measures on $C[0, 1]$ and $D[0-1]$ Bibliographical notes Bibliography List of symbols Author index Subject index as mentioned in this paper
Abstract: The Borel subsets of a metric space Probability measures in a metric space Probability measures in a metric group Probability measures in locally compact abelian groups The Kolmogorov consistency theorem and conditional probability Probability measures in a Hilbert space Probability measures on $C[0,1]$ and $D[0,1]$ Bibliographical notes Bibliography List of symbols Author index Subject index.
2,667 citations
Book•
05 Jan 1996
TL;DR: In this article, the authors consider the problem of approximating the Brownian motion by a random walk with respect to the de Moivre-laplace limit theorem and show that it is NP-hard.
Abstract: Prerequisites.- 1. The standard BRownian motion.- 1.1. The standard random walk.- 1.2. Passage times for the standard random walk.- 1.3. Hin?in's proof of the de Moivre-laplace limit theorem.- 1.4. The standard Brownian motion.- 1.5. P. Levy's construction.- 1.6. Strict Markov character.- 1.7. Passage times for the standard Brownian motion.- Note l: Homogeneous differential processes with increasing paths.- 1.8. Kolmogorov's test and the law of the iterated logarithm.- 1.9. P. Levy's Holder condition.- 1.10. Approximating the Brownian motion by a random walk.- 2. Brownian local times.- 2.1. The reflecting Brownian motion.- 2.2. P. Levy's local time.- 2.3. Elastic Brownian motion.- 2.4. t+ and down-crossings.- 2.5. T+ as Hausdorff-Besicovitch 1/2-dimensional measure.- Note 1: Submartingales.- Note 2: Hausdorff measure and dimension.- 2.6. Kac's formula for Brownian functionals.- 2.7. Bessel processes.- 2.8. Standard Brownian local time.- 2.9. BrowNian excursions.- 2.10. Application of the Bessel process to Brownian excursions.- 2.11. A time substitution.- 3. The general 1-dimensional diffusion.- 3.1. Definition.- 3.2. Markov times.- 3.3. Matching numbers.- 3.4. Singular points.- 3.5. Decomposing the general diffusion into simple pieces.- 3.6. Green operators and the space D.- 3.7. Generators.- 3.8. Generators continued.- 3.9. Stopped diffusion.- 4. Generators.- 4.1. A general view.- 4.2. G as local differential operator: conservative non-singular case.- 4.3. G as local differential operator: general non-singular case.- 4.4. A second proof.- 4.5. G at an isolated singular point.- 4.6. Solving G*u = ? u.- 4.7. G as global differential operator: non-singular case.- 4.8. G on the shunts.- 4.9. G as global differential operator: singular case.- 4.10. Passage times.- Note 1: Differential processes with increasing paths.- 4.11. Eigen-differential expansions for Green functions and transition densities.- 4.12. Kolmogorov's test.- 5. Time changes and killing.- 5.1. Construction of sample paths: a general view.- 5.2. Time changes: Q = R1.- 5.3. Time changes: Q = [0, + ?).- 5.4. Local times.- 5.5. Subordination and chain rule.- 5.6. Killing times.- 5.7. Feller's Brownian motions.- 5.8. Ikeda's example.- 5.9. Time substitutions must come from local time integrals.- 5.10. Shunts.- 5.11. Shunts with killing.- 5.12. Creation of mass.- 5.13. A parabolic equation.- 5.14. Explosions.- 5.15. A non-linear parabolic equation.- 6. Local and inverse local times.- 6.1. Local and inverse local times.- 6.2. Levy measures.- 6.3. t and the intervals of [0, + ?) - ?.- 6.4. A counter example: t and the intervals of [0, + ?) - ?.- 6.5a t and downcrossings.- 6.5b t as Hausdorff measure.- 6.5c t as diffusion.- 6.5d Excursions.- 6.6. Dimension numbers.- 6.7. Comparison tests.- Note 1: Dimension numbers and fractional dimensional capacities.- 6.8. An individual ergodic theorem.- 7. Brownian motion in several dimensions.- 7.1. Diffusion in several dimensions.- 7.2. The standard Brownian motion in several dimensions.- 7.3. Wandering out to ?.- 7.4. Greenian domains and Green functions.- 7.5. Excessive functions.- 7.6. Application to the spectrum of ?/2.- 7.7. Potentials and hitting probabilities.- 7.8. Newtonian capacities.- 7.9. Gauss's quadratic form.- 7.10. Wiener's test.- 7.11. Applications of Wiener's test.- 7.12. Dirichlet problem.- 7.13. Neumann problem.- 7.14. Space-time Brownian motion.- 7.15. Spherical Brownian motion and skew products.- 7.16. Spinning.- 7.17. An individual ergodic theorem for the standard 2-dimensional BROWNian motion.- 7.18. Covering Brownian motions.- 7.19. Diffusions with Brownian hitting probabilities.- 7.20. Right-continuous paths.- 7.21. Riesz potentials.- 8. A general view of diffusion in several dimensions.- 8.1. Similar diffusions.- 8.2. G as differential operator.- 8.3. Time substitutions.- 8.4. Potentials.- 8.5. Boundaries.- 8.6. Elliptic operators.- 8.7. Feller's little boundary and tail algebras.- List of notations.
2,063 citations