In probability theory, reflected Brownian motion (or regulated Brownian motion,[1][2] both with the acronym RBM) is a Wiener process in a space with reflecting boundaries.
For example it can describe the motion of hard spheres in water confined between two walls.
[4] RBMs have been shown to describe queueing models experiencing heavy traffic[2] as first proposed by Kingman[5] and proven by Iglehart and Whitt.
[6][7] A d–dimensional reflected Brownian motion Z is a stochastic process on
uniquely defined by where X(t) is an unconstrained Brownian motion with drift μ and variance Σ, and[9] with Y(t) a d–dimensional vector where The reflection matrix describes boundary behaviour.
the process behaves like a Wiener process; on the boundary "roughly speaking, Z is pushed in direction Rj whenever the boundary surface
"The problem of recurrence classification for SRBMs in four and higher dimensions remains open.
The heat kernel for reflected Brownian motion at
The stationary distribution of a reflected Brownian motion in multiple dimensions is tractable analytically when there is a product form stationary distribution,[10] which occurs when the process is stable and[11] where D = diag(Σ).
In this case the probability density function is[8] where ηk = 2μkγk/Σkk and γ = R−1μ.
Closed-form expressions for situations where the product form condition does not hold can be computed numerically as described below in the simulation section.
The following MATLAB program creates a sample path.
[12] The error involved in discrete simulations has been quantified.
[13] QNET allows simulation of steady state RBMs.
[14][15][16] Feller described possible boundary condition for the process[17][18][19]