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:
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
i.e. has the Rayleigh distribution with parameter 1, the same distribution as , where is an exponential random variable with parameter 1.