Brownian meander

In the mathematical theory of probability, Brownian meander $W^{+}=\{W_{t}^{+},t\in [0,1]\}$ is a continuous non-homogeneous Markov process defined as follows:

Let $W=\{W_{t},t\geq 0\}$ be a standard one-dimensional Brownian motion, and $\tau :=\sup\{t\in [0,1]:W_{t}=0\}$, i.e. the last time before t = 1 when $W$ visits $\{0\}$. Then the Brownian meander is defined by the following:

$W_{t}^{+}:={\frac {1}{\sqrt {1-\tau }}}|W_{\tau +t(1-\tau )}|,\quad t\in [0,1].$

In words, let $\tau$ be the last time before 1 that a standard Brownian motion visits $\{0\}$. ($\tau <1$ almost surely.) We snip off and discard the trajectory of Brownian motion before $\tau$, and scale the remaining part so that it spans a time interval of length 1. The scaling factor for the spatial axis must be square root of the scaling factor for the time axis. The process resulting from this snip-and-scale procedure is a Brownian meander. As the name suggests, it is a piece of Brownian motion that spends all its time away from its starting point $\{0\}$.

The transition density $p(s,x,t,y)\,dy:=P(W_{t}^{+}\in dy\mid W_{s}^{+}=x)$ of Brownian meander is described as follows:

For $0<s<t\leq 1$ and $x,y>0$, and writing

$\varphi _{t}(x):={\frac {\exp\{-x^{2}/(2t)\}}{\sqrt {2\pi t}}}\quad {\text{and}}\quad \Phi _{t}(x,y):=\int _{x}^{y}\varphi _{t}(w)\,dw,$

we have

${\begin{aligned}p(s,x,t,y)\,dy:={}&P(W_{t}^{+}\in dy\mid W_{s}^{+}=x)\\={}&{\bigl (}\varphi _{t-s}(y-x)-\varphi _{t-s}(y+x){\bigl )}{\frac {\Phi _{1-t}(0,y)}{\Phi _{1-s}(0,x)}}\,dy\end{aligned}}$

and

$p(0,0,t,y)\,dy:=P(W_{t}^{+}\in dy)=2{\sqrt {2\pi }}{\frac {y}{t}}\varphi _{t}(y)\Phi _{1-t}(0,y)\,dy.$

In particular,

$P(W_{1}^{+}\in dy)=y\exp\{-y^{2}/2\}\,dy,\quad y>0,$

i.e. $W_{1}^{+}$ has the Rayleigh distribution with parameter 1, the same distribution as ${\sqrt {2\mathbf {e} }}$, where $\mathbf {e}$ is an exponential random variable with parameter 1.

References

• Durett, Richard; Iglehart, Donald; Miller, Douglas (1977). "Weak convergence to Brownian meander and Brownian excursion". The Annals of Probability. 5 (1): 117–129. doi:10.1214/aop/1176995895.
• Revuz, Daniel; Yor, Marc (1999). Continuous Martingales and Brownian Motion (2nd ed.). New York: Springer-Verlag. ISBN 3-540-57622-3.