# 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

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

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

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

we have

\begin{align} p(s,x,t,y) \, dy &:= P(W^+_t \in dy \mid W^+_s = x) \\ &= \bigl( \phi_{t-s}(y-x) - \phi_{t-s}(y+x) \bigl) \frac{\Phi_{1-t}(0,y)}{\Phi_{1-s}(0,x)} \, dy \end{align}

and

$p(0,0,t,y) \, dy := P(W^+_t \in dy ) = 2\sqrt{2 \pi} \frac{y}{t}\phi_t(y)\Phi_{1-t}(0,y) \, dy.$

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.