Reflected Brownian motion

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.^[3] 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.^[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]

Definition[edit]

A d–dimensional reflected Brownian motion Z is a stochastic process on ${\displaystyle \mathbb {R} _{+}^{d}}$ uniquely defined by

• a d–dimensional drift vector μ
• a d×d non-singular covariance matrix Σ and
• a d×d reflection matrix R.^[8]

where X(t) is an unconstrained Brownian motion with drift μ and variance Σ, and^[9]

${\displaystyle Z(t)=X(t)+RY(t)}$

with Y(t) a d–dimensional vector where

• Y is continuous and non–decreasing with Y(0) = 0
• Y_j only increases at times for which Z_j = 0 for j = 1,2,...,d
• Z(t) ∈ ${\displaystyle \mathbb {R} _{+}^{d}}$, t ≥ 0.

The reflection matrix describes boundary behaviour. In the interior of ${\displaystyle \scriptstyle \mathbb {R} _{+}^{d}}$ the process behaves like a Wiener process; on the boundary "roughly speaking, Z is pushed in direction R^j whenever the boundary surface ${\displaystyle \scriptstyle \{z\in \mathbb {R} _{+}^{d}:z_{j}=0\}}$ is hit, where R^j is the jth column of the matrix R."^[9] The process Y_j is the local time of the process on the corresponding section of the boundary.

Stability conditions[edit]

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."^[9] In the special case where R is an M-matrix then necessary and sufficient conditions for stability are^[9]

1. R is a non-singular matrix and
2. R⁻¹μ < 0.

Marginal and stationary distribution[edit]

One dimension[edit]

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

${\displaystyle \mathbb {P} (Z(t)\leq z)=\Phi \left({\frac {z-\mu t}{\sigma t^{1/2}}}\right)-e^{2\mu z/\sigma ^{2}}\Phi \left({\frac {-z-\mu t}{\sigma t^{1/2}}}\right)}$

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^[2]

${\displaystyle \mathbb {P} (Z<z)=1-e^{2\mu z/\sigma ^{2}}.}$

For fixed t, the distribution of Z(t) coincides with the distribution of the running maximum M(t) of the Brownian motion,

${\displaystyle Z(t)\sim M(t)=\sup _{s\in [0,t]}X(s).}$

But be aware that the distributions of the processes as a whole are very different. In particular, M(t) is increasing in t, which is not the case for Z(t).

The heat kernel for reflected Brownian motion at ${\displaystyle p_{b}}$:

${\displaystyle f(x,p_{b})={\frac {e^{-((x-u)/a)^{2}/2}+e^{-((x+u-2p_{b})/a)^{2}/2}}{a(2\pi )^{1/2}}}}$

For the plane above ${\displaystyle x\geq p_{b}}$

Multiple dimensions[edit]

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]

${\displaystyle 2\Sigma =RD+DR'}$

where D = diag(Σ). In this case the probability density function is^[8]

${\displaystyle p(z_{1},z_{2},\ldots ,z_{d})=\prod _{k=1}^{d}\eta _{k}e^{-\eta _{k}z_{k}}}$

where η_k = 2μ_kγ_k/Σ_kk 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.

Simulation[edit]

One dimension[edit]

In one dimension the simulated process is the absolute value of a Wiener process. The following MATLAB program creates a sample path.^[12]

% rbm.m n = 10^4; h=10^(-3); t=h.*(0:n); mu=-1; X = zeros(1, n+1); M=X; B=X; B(1)=3; X(1)=3; for k=2:n+1     Y = sqrt(h) * randn; U = rand(1);     B(k) = B(k-1) + mu * h - Y;     M = (Y + sqrt(Y ^ 2 - 2 * h * log(U))) / 2;     X(k) = max(M-Y, X(k-1) + h * mu - Y); end subplot(2, 1, 1) plot(t, X, 'k-'); subplot(2, 1, 2) plot(t, X-B, 'k-'); 

The error involved in discrete simulations has been quantified.^[13]

Multiple dimensions[edit]

QNET allows simulation of steady state RBMs.^[14][15][16]

Other boundary conditions[edit]

Feller described possible boundary condition for the process^[17][18][19]

• absorption^[17] or killed Brownian motion,^[20] a Dirichlet boundary condition
• instantaneous reflection,^[17] as described above a Neumann boundary condition
• elastic reflection, a Robin boundary condition
• delayed reflection^[17] (the time spent on the boundary is positive with probability one)
• partial reflection^[17] where the process is either immediately reflected or is absorbed
• sticky Brownian motion.^[21]

See also[edit]

• Skorokhod problem

References[edit]