The Library
Consistent families of Brownian motions and stochastic flows of kernels
Tools
Howitt, Chris and Warren, Jon. (2009) Consistent families of Brownian motions and stochastic flows of kernels. The Annals of Probability, Vol.37 (No.4). pp. 12371272. ISSN 00911798

PDF
WRAP_warren_euclid.aop.1248182138.pdf  Published Version  Requires a PDF viewer such as GSview, Xpdf or Adobe Acrobat Reader Download (342Kb) 
Official URL: http://dx.doi.org/10.1214/08AOP431
Abstract
Consider the following mechanism for the random evolution of a distribution of mass on the integer lattice Z. At unit rate, independently for each site, the mass at the site is split into two parts by choosing a random proportion distributed according to some specified probability measure on [0, 1] and dividing the mass in that proportion. One part then moves to each of the two adjacent sites. This paper considers a continuous analogue of this evolution, which may be described by means of a stochastic flow of kernels, the theory of which was developed by Le Jan and Raimond. One of their results is that such a flow is characterized by specifying its N point motions, which form a consistent family of Brownian motions. This means for each dimension N we have a diffusion in RN, whose N coordinates are all Brownian motions. Any M coordinates taken from the Ndimensional process are distributed as the Mdimensional process in the family. Moreover, in this setting, the only interactions between coordinates are local: when coordinates differ in value they evolve independently of each other. In this paper we explain how such multidimensional diffusions may be constructed and characterized via martingale problems.
[error in script] [error in script]Item Type:  Journal Article 

Subjects:  Q Science > QA Mathematics 
Divisions:  Faculty of Science > Statistics 
Library of Congress Subject Headings (LCSH):  Brownian motion processes, Kernel functions, Stochastic processes, Diffusion processes 
Journal or Publication Title:  The Annals of Probability 
Publisher:  Institute of Mathematical Statistics 
ISSN:  00911798 
Date:  2009 
Volume:  Vol.37 
Number:  No.4 
Page Range:  pp. 12371272 
Identification Number:  10.1214/08AOP431 
Status:  Peer Reviewed 
Access rights to Published version:  Restricted or Subscription Access 
References:  [1] AMIR, M. (1991). Sticky Brownian motion as the strong limit of a sequence of random walks. Stochastic Process. Appl. 39 221–237. MR1136247 [2] BORODIN, A. N. and SALMINEN, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd ed. Birkhäuser, Basel. MR1912205 [3] FALKOVICH, G., GAW¸EDZKI, K. and VERGASSOLA, M. (2001). Particles and fields in fluid turbulence. Rev. Modern Phys. 73 913–975. MR1878800 [4] GAW¸EDZKI, K. and HORVAI, P. (2004). Sticky behavior of fluid particles in the compressible Kraichnan model. J. Statist. Phys. 116 1247–1300. MR2096036 [5] HARRISON, J. M. and LEMOINE, A. J. (1981). Sticky Brownian motion as the limit of storage processes. J. Appl. Probab. 18 216–226. MR598937 [6] HOWITT, C. J. andWARREN, J. (2008). Dynamics for the Brownian web and the erosion flow. Stochastic Process. Appl. To appear. [7] IKEDA, N. andWATANABE, S. (1973). The local structure of a class of diffusions and related problems. In Proceedings of the Second Japan–USSR Symposium on Probability Theory (Kyoto, 1972). Lecture Notes in Math. 330 124–169. Springer, Berlin. MR0451425 [8] JACOD, J. and SHIRYAEV, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin. MR1943877 [9] LE JAN, Y. and LEMAIRE, S. (2004). Products of Beta matrices and sticky flows. Probab. Theory Related Fields 130 109–134. MR2092875 [10] LE JAN, Y. and RAIMOND, O. (2004). Flows, coalescence and noise. Ann. Probab. 32 1247– 1315. MR2060298 [11] LE JAN, Y. and RAIMOND, O. (2004). Sticky flows on the circle and their noises. Probab. Theory Related Fields 129 63–82. MR2052863 [12] PROTTER, P. E. (2005). Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability 21. Springer, Berlin. MR2273672 [13] REVUZ, D. and YOR, M. (1999). Continuous Martingales and Brownian Motion, 3rd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 293. Springer, Berlin. MR1725357 [14] STROOCK, D. W. and VARADHAN, S. R. S. (1979). Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 233. Springer, Berlin. MR532498 [15] SUN, R. and SWART, J. M. (2008). The Brownian net. Ann. Probab. 36 1153–1208. [16] TSIRELSON, B. (2004). Nonclassical stochastic flows and continuous products. Probab. Surv. 1 173–298 (electronic). MR2068474 [17] WARREN, J. (1997). Branching processes, the Ray–Knight theorem, and sticky Brownian motion. In Séminaire de Probabilités, XXXI. Lecture Notes in Math. 1655 1–15. Springer, Berlin. MR1478711 
URI:  http://wrap.warwick.ac.uk/id/eprint/37376 
Data sourced from Thomson Reuters' Web of Knowledge
Actions (login required)
View Item 