In the mathematical theory of probability, Brownian meander W + = { W t + , t ∈ [ 0 , 1 ] } is a continuous non-homogeneous Markov process defined as follows:
Let W = { W t , t ≥ 0 } be a standard one-dimensional Brownian motion, and τ := sup { t ∈ [ 0 , 1 ] : W t = 0 } , i.e. the last time before t = 1 when W visits { 0 } . Then
W t + := 1 1 − τ | W τ + t ( 1 − τ ) | , t ∈ [ 0 , 1 ] . The transition density p ( s , x , t , y ) d y := P ( W t + ∈ d y ∣ W s + = x ) of Brownian meander is described as follows:
For 0 < s < t ≤ 1 and x , y > 0 , and writing
ϕ t ( x ) := exp { − x 2 / ( 2 t ) } 2 π t and Φ t ( x , y ) := ∫ x y ϕ t ( w ) d w , we have
p ( s , x , t , y ) d y := P ( W t + ∈ d y ∣ W s + = x ) = ( ϕ t − s ( y − x ) − ϕ t − s ( y + x ) ) Φ 1 − t ( 0 , y ) Φ 1 − s ( 0 , x ) d y and
p ( 0 , 0 , t , y ) d y := P ( W t + ∈ d y ) = 2 2 π y t ϕ t ( y ) Φ 1 − t ( 0 , y ) d y . In particular,
P ( W 1 + ∈ d y ) = y exp { − y 2 / 2 } d y , y > 0 , i.e. W 1 + has the Rayleigh distribution with parameter 1, the same distribution as 2 e , where e is an exponential random variable with parameter 1.
