Brownian meander

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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 transition density of Brownian meander is described as follows:

For and , and writing

we have


In particular,

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