In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...
, reflected Brownian motion (or regulated Brownian motion,
both with the acronym RBM) is a
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
in a space with reflecting boundaries. In the
physical
Physical may refer to:
*Physical examination
In a physical examination, medical examination, or clinical examination, a medical practitioner examines a patient for any possible medical signs or symptoms of a medical condition. It generally co ...
literature, this process describes
diffusion
Diffusion is the net movement of anything (for example, atoms, ions, molecules, energy) generally from a region of higher concentration to a region of lower concentration. Diffusion is driven by a gradient in Gibbs free energy or chemical p ...
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 model
Queueing theory is the mathematical study of waiting lines, or queues. A queueing model is constructed so that queue lengths and waiting time can be predicted. Queueing theory is generally considered a branch of operations research because the ...
s experiencing
heavy traffic
''Heavy Traffic'' is a 1973 American live-action/animated drama film written and directed by Ralph Bakshi. The film, which begins, ends, and occasionally combines with live-action, explores the often surreal fantasies of a young New York City ...
as first proposed by
Kingman and proven by Iglehart and
Whitt
Whitt is a surname. It may refer to:
* Brandon Whitt (1982– ), American racing driver
* Cole Whitt (1991– ), American racing driver
* Don Whitt (1930–2013), an American professional golfer
* Ernie Whitt
Leo Ernest Whitt (born June 13, 1 ...
.
Definition
A ''d''–dimensional reflected Brownian motion ''Z'' is a
stochastic process
In probability theory and related fields, a stochastic () or random process is a mathematical object usually defined as a family of random variables. Stochastic processes are widely used as mathematical models of systems and phenomena that appea ...
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
Brownian motion, or pedesis (from grc, πήδησις "leaping"), is the random motion of particles suspended in a medium (a liquid or a gas).
This pattern of motion typically consists of random fluctuations in a particle's position insi ...
and
::
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'') ∈
, t ≥ 0.
The reflection matrix describes boundary behaviour. In the interior of
the process behaves like a
Wiener process
In mathematics, the Wiener process is a real-valued continuous-time stochastic process named in honor of American mathematician Norbert Wiener for his investigations on the mathematical properties of the one-dimensional Brownian motion. It is o ...
; on the boundary "roughly speaking, ''Z'' is pushed in direction ''R''
''j'' whenever the boundary surface
is hit, where ''R''
''j'' is the ''j''th column of the matrix ''R''."
Stability conditions
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 In mathematics, especially linear algebra, an ''M''-matrix is a ''Z''-matrix with eigenvalues whose real parts are nonnegative. The set of non-singular ''M''-matrices are a subset of the class of ''P''-matrices, and also of the class of inverse-p ...
then necessary and sufficient conditions for stability are
# ''R'' is a
non-singular matrix
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that
:\mathbf = \mathbf = \mathbf_n \
where denotes the -by- identity matrix and the multiplicati ...
and
# ''R''
−1''μ'' < 0.
Marginal and stationary distribution
One dimension
The
marginal distribution
In probability theory and statistics, the marginal distribution of a subset of a collection of random variables is the probability distribution of the variables contained in the subset. It gives the probabilities of various values of the variables ...
(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 ''σ''
2 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
In probability theory and statistics, the exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average ...
::