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 $W=\{W_t,t\ge 0\}$ 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:

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

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

${\phi }_t(x):=\frac{\exp \{-x^2/(2t)\}}{\sqrt{2\pi t}}\text{and}{\mathrm{\Phi }}_t(x,y):=\underset{x}{\overset{y}{\int }}{\phi }_t(w)dw,$

we have

$\begin{array}{cc}p(s,x,t,y)dy:=& P(W_t^{+}\in dy\mid W_s^{+}=x)\\ =& ({\phi }_{t-s}(y-x)-{\phi }_{t-s}(y+x))\frac{{\mathrm{\Phi }}_{1-t}(0,y)}{{\mathrm{\Phi }}_{1-s}(0,x)}dy\end{array}$

and

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

In particular,

$P(W_1^{+}\in dy)=y\exp \{-y^2/2\}dy,y>0,$

i.e. $W_1^{+}$ has the Rayleigh distribution with parameter 1, the same distribution as $\sqrt{2𝐞}$, where $𝐞$ is an exponential random variable with parameter 1.

References

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