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
Definition
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.[8]
where X(t) is an unconstrained Brownian motion with drift μ and variance Σ, and[9]
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."[9] The process Yj is the local time of the process on the corresponding section of the boundary.
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."[9] In the special case where R is an M-matrix then necessary and sufficient conditions for stability are[9]
- R is a non-singular matrixand
- R−1μ < 0.
Marginal and stationary distribution
One dimension
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
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. In particular, M(t) is increasing in t, which is not the case for Z(t).
The heat kernel for reflected Brownian motion at :
For the plane above
Multiple dimensions
The stationary distribution of a reflected Brownian motion in multiple dimensions is tractable analytically when there is a
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.
Simulation
One dimension
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
QNET allows simulation of steady state RBMs.[14][15][16]
Other boundary conditions
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
References
- ISBN 9780470400531.
- ^ ISBN 978-0471819394.
- S2CID 121673717.
- ISSN 1063-651X.
- JSTOR 2984229.
- S2CID 202104090.
- S2CID 120281300. Retrieved 30 Nov 2012.
- ^ .
- ^ S2CID 2251853.
- JSTOR 2959751.
- doi:10.1137/0141030.
- ISBN 978-1118014950.
- JSTOR 2245096.
- JSTOR 2959623.
- ^ Dai, Jiangang "Jim" (1990). "Section A.5 (code for BNET)" (PDF). Steady-state analysis of reflected Brownian motions: characterization, numerical methods and queueing applications (Ph. D. thesis) (Thesis). Stanford University. Dept. of Mathematics. Retrieved 5 December 2012.
- JSTOR 2959654.
- ^ doi:10.1137/1107002.
- MR 0063607.
- ^ Engelbert, H. J.; Peskir, G. (2012). "Stochastic Differential Equations for Sticky Brownian Motion" (PDF). Probab. Statist. Group Manchester Research Report (5).
- ISBN 978-3-642-63381-2.
- ISBN 978-3-540-60629-1.