Reflected Brownian motion

In probability theory, reflected Brownian motion (or regulated Brownian motion, both with the acronym RBM) is a Wiener process in a space with reflecting boundaries. RBMs have been shown to describe queueing models experiencing heavy traffic as first proposed by Kingman and proven by Iglehart and Whitt. A d–dimensional reflected Brownian motion Z is a stochastic process on R + d {displaystyle mathbb {R} _{+}^{d}} uniquely defined by where X(t) is an unconstrained Brownian motion and with Y(t) a d–dimensional vector where The reflection matrix describes boundary behaviour. In the interior of R + d {displaystyle scriptstyle mathbb {R} _{+}^{d}} the process behaves like a Wiener process, on the boundary 'roughly speaking, Z is pushed in direction Rj whenever the boundary surface { z ∈ R + d : z j = 0 } {displaystyle scriptstyle {zin mathbb {R} _{+}^{d}:z_{j}=0}} is hit, where Rj is the jth column of the matrix R.' Stability conditions are known for RBMs in 1, 2, and 3 dimensions. 'The problem of recurrence classification for SRBMs in four and higher dimensions remains open.' In the special case where R is an M-matrix then necessary and sufficient conditions for stability are The marginal distribution (transient distribution) of a one-dimensional Brownian motion starting at 0 restricted to positive values (a single reflecting barrier at 0) with drift μ and variance σ2 is