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


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

we have



\begin{align} 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{align} $$

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.