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Stochastic flows in the Brownian web and net
About this Title
Emmanuel Schertzer, 109 Montague Street, Brooklyn, New York, New York 11201, Rongfeng Sun, Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076, Singapore and Jan M. Swart, Institute of Information Theory and Automation of the ASCR (ÚTIA), Pod vodárenskou věží 4, 18208 Praha 8, Czech Republic
Publication: Memoirs of the American Mathematical Society
Publication Year:
2014; Volume 227, Number 1065
ISBNs: 978-0-8218-9088-2 (print); 978-1-4704-1426-9 (online)
DOI: https://doi.org/10.1090/S0065-9266-2013-00687-9
Published electronically: May 24, 2013
Keywords: Brownian web,
Brownian net,
stochastic flow of kernels,
measure-valued process,
Howitt-Warren flow,
linear system,
random walk in random environment,
finite graph representation
MSC: Primary 82C21; Secondary 60K35, 60K37, 60D05
Table of Contents
Chapters
- 1. Introduction
- 2. Results for Howitt-Warren flows
- 3. Construction of Howitt-Warren flows in the Brownian web
- 4. Construction of Howitt-Warren flows in the Brownian net
- 5. Outline of the proofs
- 6. Coupling of the Brownian web and net
- 7. Construction and convergence of Howitt-Warren flows
- 8. Support properties
- 9. Atomic or non-atomic
- 10. Infinite starting mass and discrete approximation
- 11. Ergodic properties
- A. The Howitt-Warren martingale problem
- B. The Hausdorff topology
- C. Some measurability issues
- D. Thinning and Poissonization
- E. A one-sided version of Kolmogorov’s moment criterion
Abstract
It is known that certain one-dimensional nearest-neighbor random walks in i.i.d. random space-time environments have diffusive scaling limits. Here, in the continuum limit, the random environment is represented by a ‘stochastic flow of kernels’, which is a collection of random kernels that can be loosely interpreted as the transition probabilities of a Markov process in a random environment. The theory of stochastic flows of kernels was first developed by Le Jan and Raimond, who showed that each such flow is characterized by its $n$-point motions. Our work focuses on a class of stochastic flows of kernels with Brownian $n$-point motions which, after their inventors, will be called Howitt-Warren flows.
Our main result gives a graphical construction of general Howitt-Warren flows, where the underlying random environment takes on the form of a suitably marked Brownian web. This extends earlier work of Howitt and Warren who showed that a special case, the so-called ‘erosion flow’, can be constructed from two coupled ‘sticky Brownian webs’. Our construction for general Howitt-Warren flows is based on a Poisson marking procedure developed by Newman, Ravishankar and Schertzer for the Brownian web. Alternatively, we show that a special subclass of the Howitt-Warren flows can be constructed as random flows of mass in a Brownian net, introduced by Sun and Swart.
Using these constructions, we prove some new results for the Howitt-Warren flows. In particular, we show that the kernels spread with a finite speed and have a locally finite support at deterministic times if and only if the flow is embeddable in a Brownian net. We show that the kernels are always purely atomic at deterministic times, but, with the exception of the erosion flows, exhibit random times when the kernels are purely non-atomic. We moreover prove ergodic statements for a class of measure-valued processes induced by the Howitt-Warren flows.
Our work also yields some new results in the theory of the Brownian web and net. In particular, we prove several new results about coupled sticky Brownian webs and about a natural coupling of a Brownian web with a Brownian net. We also introduce a ‘finite graph representation’ which gives a precise description of how paths in the Brownian net move between deterministic times.
- Richard Alejandro Arratia, COALESCING BROWNIAN MOTIONS ON THE LINE, ProQuest LLC, Ann Arbor, MI, 1979. Thesis (Ph.D.)–The University of Wisconsin - Madison. MR 2630231
- R. Arratia. Coalescing Brownian motions and the voter model on $\mathbb {Z}$. Unpublished partial manuscript. Available from rarratia@math.usc.edu.
- David Barbato, FKG inequality for Brownian motion and stochastic differential equations, Electron. Comm. Probab. 10 (2005), 7–16. MR 2119149, DOI 10.1214/ECP.v10-1127
- Patrick Billingsley, Convergence of probability measures, 2nd ed., Wiley Series in Probability and Statistics: Probability and Statistics, John Wiley & Sons, Inc., New York, 1999. A Wiley-Interscience Publication. MR 1700749
- N. Bourbaki, Éléments de mathématique. Fasc. XXXVIII: Groupes et algèbres de Lie. Chapitre VII: Sous-algèbres de Cartan, éléments réguliers. Chapitre VIII: Algèbres de Lie semi-simples déployées, Actualités Scientifiques et Industrielles, No. 1364. Hermann, Paris, 1975 (French). MR 0453824
- M. T. Barlow and M. Yor, (Semi-) martingale inequalities and local times, Z. Wahrsch. Verw. Gebiete 55 (1981), no. 3, 237–254. MR 608019, DOI 10.1007/BF00532117
- Donald A. Dawson, Measure-valued Markov processes, École d’Été de Probabilités de Saint-Flour XXI—1991, Lecture Notes in Math., vol. 1541, Springer, Berlin, 1993, pp. 1–260. MR 1242575, DOI 10.1007/BFb0084190
- Richard Durrett, Probability: theory and examples, 2nd ed., Duxbury Press, Belmont, CA, 1996. MR 1609153
- Stewart N. Ethier and Thomas G. Kurtz, Markov processes, Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics, John Wiley & Sons, Inc., New York, 1986. Characterization and convergence. MR 838085
- S.N. Evans, B. Morris and A. Sen. Coalescing systems of non-Brownian particles, Probab. Theory Related Fields 156 (2013), 307–342. .
- William Feller, An introduction to probability theory and its applications. Vol. II, John Wiley & Sons, Inc., New York-London-Sydney, 1966. MR 0210154
- L. R. G. Fontes, M. Isopi, and C. M. Newman, Random walks with strongly inhomogeneous rates and singular diffusions: convergence, localization and aging in one dimension, Ann. Probab. 30 (2002), no. 2, 579–604. MR 1905852, DOI 10.1214/aop/1023481003
- L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, The Brownian web, Proc. Natl. Acad. Sci. USA 99 (2002), no. 25, 15888–15893. MR 1944976, DOI 10.1073/pnas.252619099
- L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, The Brownian web: characterization and convergence, Ann. Probab. 32 (2004), no. 4, 2857–2883. MR 2094432, DOI 10.1214/009117904000000568
- L. R. G. Fontes, M. Isopi, C. M. Newman, and K. Ravishankar, Coarsening, nucleation, and the marked Brownian web, Ann. Inst. H. Poincaré Probab. Statist. 42 (2006), no. 1, 37–60 (English, with English and French summaries). MR 2196970, DOI 10.1016/j.anihpb.2005.01.003
- J. Henrikson. Completeness and total boundedness of the Hausdorff metric. MIT Undergraduate Journal of Mathematics 1 Number 1, 69–79, 1999.
- Wassily Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. Assoc. 58 (1963), 13–30. MR 144363
- Chris Howitt and Jon Warren, Consistent families of Brownian motions and stochastic flows of kernels, Ann. Probab. 37 (2009), no. 4, 1237–1272. MR 2546745, DOI 10.1214/08-AOP431
- Chris Howitt and Jon Warren, Dynamics for the Brownian web and the erosion flow, Stochastic Process. Appl. 119 (2009), no. 6, 2028–2051. MR 2519355, DOI 10.1016/j.spa.2008.10.003
- A.St. John and H. Mathur. Correlations and critical behavior of the $q$ model. Phys. Rev. E 84, 051303, 2011.
- Olav Kallenberg, Foundations of modern probability, 2nd ed., Probability and its Applications (New York), Springer-Verlag, New York, 2002. MR 1876169
- Ioannis Karatzas and Steven E. Shreve, Brownian motion and stochastic calculus, 2nd ed., Graduate Texts in Mathematics, vol. 113, Springer-Verlag, New York, 1991. MR 1121940
- Thomas G. Kurtz, Martingale problems for conditional distributions of Markov processes, Electron. J. Probab. 3 (1998), no. 9, 29. MR 1637085, DOI 10.1214/EJP.v3-31
- Y. Le Jan and S. Lemaire, Products of Beta matrices and sticky flows, Probab. Theory Related Fields 130 (2004), no. 1, 109–134. MR 2092875, DOI 10.1007/s00440-004-0358-7
- Yves Le Jan and Olivier Raimond, Flows, coalescence and noise, Ann. Probab. 32 (2004), no. 2, 1247–1315. MR 2060298, DOI 10.1214/009117904000000207
- Yves Le Jan and Olivier Raimond, Sticky flows on the circle and their noises, Probab. Theory Related Fields 129 (2004), no. 1, 63–82. MR 2052863, DOI 10.1007/s00440-003-0324-9
- M. Lewandowska, H. Mathur, and Y.-K. Yu. Dynamics and critical behavior of the $q$ model. Phys. Rev. E 64, 026107, 2001.
- Thomas M. Liggett, A characterization of the invariant measures for an infinite particle system with interactions, Trans. Amer. Math. Soc. 179 (1973), 433–453. MR 326867, DOI 10.1090/S0002-9947-1973-0326867-1
- Thomas M. Liggett and Frank Spitzer, Ergodic theorems for coupled random walks and other systems with locally interacting components, Z. Wahrsch. Verw. Gebiete 56 (1981), no. 4, 443–468. MR 621659, DOI 10.1007/BF00531427
- Thomas M. Liggett, Interacting particle systems, Classics in Mathematics, Springer-Verlag, Berlin, 2005. Reprint of the 1985 original. MR 2108619
- Klaus Matthes, Johannes Kerstan, and Joseph Mecke, Infinitely divisible point processes, John Wiley & Sons, Chichester-New York-Brisbane, 1978. Translated from the German by B. Simon; Wiley Series in Probability and Mathematical Statistics. MR 0517931
- J.R. Munkres. Topology, 2nd ed. Prentice Hall, Upper Saddle River, 2000.
- C. M. Newman, K. Ravishankar, and E. Schertzer, Marking $(1,2)$ points of the Brownian web and applications, Ann. Inst. Henri Poincaré Probab. Stat. 46 (2010), no. 2, 537–574 (English, with English and French summaries). MR 2667709, DOI 10.1214/09-AIHP325
- Sidney I. Resnick, Extreme values, regular variation, and point processes, Applied Probability. A Series of the Applied Probability Trust, vol. 4, Springer-Verlag, New York, 1987. MR 900810
- L. C. G. Rogers and J. W. Pitman, Markov functions, Ann. Probab. 9 (1981), no. 4, 573–582. MR 624684
- Rongfeng Sun and Jan M. Swart, The Brownian net, Ann. Probab. 36 (2008), no. 3, 1153–1208. MR 2408586, DOI 10.1214/07-AOP357
- Emmanuel Schertzer, Rongfeng Sun, and Jan M. Swart, Special points of the Brownian net, Electron. J. Probab. 14 (2009), 805–864. MR 2497454, DOI 10.1214/EJP.v14-641
- Charles Stone, On local and ratio limit theorems, Proc. Fifth Berkeley Sympos. Math. Statist. and Probability (Berkeley, Calif., 1965/66) Univ. California Press, Berkeley, Calif., 1967, pp. 217–224. MR 0222939
- Florin Soucaliuc, Bálint Tóth, and Wendelin Werner, Reflection and coalescence between independent one-dimensional Brownian paths, Ann. Inst. H. Poincaré Probab. Statist. 36 (2000), no. 4, 509–545 (English, with English and French summaries). MR 1785393, DOI 10.1016/S0246-0203(00)00136-9
- Bálint Tóth and Wendelin Werner, The true self-repelling motion, Probab. Theory Related Fields 111 (1998), no. 3, 375–452. MR 1640799, DOI 10.1007/s004400050172