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. In the physical literature, this process describes diffusion in a confined space and it is often called confined Brownian motion. For example it can describe the motion of hard spheres in water confined between two walls. 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 uniquely defined by a d–dimensional drift vector μ a d×d non-singular covariance matrix Σ and a d×d reflection matrix R. where X(t) is an unconstrained Brownian motion and with Y(t) a d–dimensional vector where Y is continuous and non–decreasing with Y(0) = 0 Yj only increases at times for which Zj = 0 for j = 1,2,...,d Z(t) ∈ , t ≥ 0. The reflection matrix describes boundary behaviour. In the interior of the process behaves like a Wiener process; on the boundary "roughly speaking, Z is pushed in direction Rj whenever the boundary surface 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 R is a non-singular matrix and R−1μ < 0. 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 for all t ≥ 0, (with Φ the cumulative distribution function of the normal distribution) which yields (for μ < 0) when taking t → ∞ an exponential distribution For fixed t, the distribution of Z(t) coincides with the distribution of the running maximum M(t) of the Brownian motion, But be aware that the distributions of the processes as a whole are very different.