The Annals of Applied Probability

Multidimensional sticky Brownian motions as limits of exclusion processes

Miklós Z. Rácz and Mykhaylo Shkolnikov

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Abstract

We study exclusion processes on the integer lattice in which particles change their velocities due to stickiness. Specifically, whenever two or more particles occupy adjacent sites, they stick together for an extended period of time, and the entire particle system is slowed down until the “collision” is resolved. We show that under diffusive scaling of space and time such processes converge to what one might refer to as a sticky reflected Brownian motion in the wedge. The latter behaves as a Brownian motion with constant drift vector and diffusion matrix in the interior of the wedge, and reflects at the boundary of the wedge after spending an instant of time there. In particular, this leads to a natural multidimensional generalization of sticky Brownian motion on the half-line, which is of interest in both queuing theory and stochastic portfolio theory. For instance, this can model a market, which experiences a slowdown due to a major event (such as a court trial between some of the largest firms in the market) deciding about the new market leader.

Article information

Source
Ann. Appl. Probab. Volume 25, Number 3 (2015), 1155-1188.

Dates
First available in Project Euclid: 23 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.aoap/1427124126

Digital Object Identifier
doi:10.1214/14-AAP1019

Mathematical Reviews number (MathSciNet)
MR3325271

Zentralblatt MATH identifier
1315.60111

Subjects
Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 60K35: Interacting random processes; statistical mechanics type models; percolation theory [See also 82B43, 82C43] 60H10: Stochastic ordinary differential equations [See also 34F05]

Keywords
Exclusion processes with speed change reflected Brownian motion scaling limits sticky Brownian motion stochastic differential equations stochastic portfolio theory

Citation

Rácz, Miklós Z.; Shkolnikov, Mykhaylo. Multidimensional sticky Brownian motions as limits of exclusion processes. Ann. Appl. Probab. 25 (2015), no. 3, 1155--1188. doi:10.1214/14-AAP1019. https://projecteuclid.org/euclid.aoap/1427124126


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