Process in mathematical probability theory
In the mathematical theory of probability, Brownian meander
is a continuous non-homogeneous Markov process defined as follows:
Let
be a standard one-dimensional Brownian motion, and
, i.e. the last time before t = 1 when
visits
. Then the Brownian meander is defined by the following:
![{\displaystyle W_{t}^{+}:={\frac {1}{\sqrt {1-\tau }}}|W_{\tau +t(1-\tau )}|,\quad t\in [0,1].}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7295bb216c3bb6b37935ae29a3ca157bcc1e609f)
In words, let
be the last time before 1 that a standard Brownian motion visits
. (
almost surely.) We snip off and discard the trajectory of Brownian motion before
, 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
.
The transition density
of Brownian meander is described as follows:
For
and
, and writing
![{\displaystyle \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,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fe61cf9b892170054ae31a78edc40bf70ddcce75)
we have
![{\displaystyle {\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}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/933dc455384d52dd600ffb5c0181537bf335e483)
and
![{\displaystyle 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.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91740bffb4a74f99e36db09a571fad8ee12c12c0)
In particular,
![{\displaystyle P(W_{1}^{+}\in dy)=y\exp\{-y^{2}/2\}\,dy,\quad y>0,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/20b11c708de5cbd7b280ec0ebfd2fe727b4122fa)
i.e.
has the Rayleigh distribution with parameter 1, the same distribution as
, where
is an exponential random variable with parameter 1.