Source: Ann. Probab. Volume 37, Number 4
(2009), 1237-1272.
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 N-dimensional process are distributed as the M-dimensional 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.
References
[1] Amir, M. (1991). Sticky Brownian motion as the strong limit of a sequence of random walks. Stochastic Process. Appl. 39 221–237.
[2] Borodin, A. N. and Salminen, P. (2002). Handbook of Brownian Motion—Facts and Formulae, 2nd ed. Birkhäuser, Basel.
[3] Falkovich, G., Gawȩdzki, K. and Vergassola, M. (2001). Particles and fields in fluid turbulence. Rev. Modern Phys. 73 913–975.
[4] Gawȩdzki, K. and Horvai, P. (2004). Sticky behavior of fluid particles in the compressible Kraichnan model. J. Statist. Phys. 116 1247–1300.
[5] Harrison, J. M. and Lemoine, A. J. (1981). Sticky Brownian motion as the limit of storage processes. J. Appl. Probab. 18 216–226.
Mathematical Reviews (MathSciNet):
MR598937
[6] Howitt, C. J. and Warren, J. (2008). Dynamics for the Brownian web and the erosion flow. Stochastic Process. Appl. To appear.
[7] Ikeda, N. and Watanabe, 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.
Mathematical Reviews (MathSciNet):
MR451425
[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.
[9] Le Jan, Y. and Lemaire, S. (2004). Products of Beta matrices and sticky flows. Probab. Theory Related Fields 130 109–134.
[10] Le Jan, Y. and Raimond, O. (2004). Flows, coalescence and noise. Ann. Probab. 32 1247–1315.
[11] Le Jan, Y. and Raimond, O. (2004). Sticky flows on the circle and their noises. Probab. Theory Related Fields 129 63–82.
[12] Protter, P. E. (2005). Stochastic Integration and Differential Equations. Stochastic Modelling and Applied Probability 21. Springer, Berlin.
[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.
[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.
Mathematical Reviews (MathSciNet):
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).
[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.
