Brownian excursion the Airy function and formula_22 is the confluent hypergeometric function. Janson and Louchard (2007) show that and They also give higher-order expansions in both cases. Janson (2007) gives moments of formula_19 and many other area functionals. In particular, Brownian excursions also arise in connection with queuing problems, railway traffic, and the heights of random rooted binary trees. Brownian excursion In probability theory a Brownian excursion process is a stochastic process that is closely related to a Wiener process (or Brownian motion). Realisations of Brownian excursion processes are essentially just realizations of a Wiener process selected to satisfy certain conditions. In
Schramm–Loewner evolution evolution is a probability measure on planar curves, given as the image of Wiener measure under this map. In general the curve γ need not be simple, and the domain "D" is not the complement of γ([0,"t"]) in "D", but is instead the unbounded component of the complement. There are two versions of SLE, using two families of curves, each depending on a non-negative real parameter κ: SLE depends on a choice of Brownian motion on the boundary of the domain, and there are several variations depending on what sort of Brownian motion is used: for example it might start