In mathematics, a Bessel process, named after Friedrich Bessel. The n-dimensional Bessel process is the solution to the stochastic differential equation (SDE)
where W is a 1-dimensional Wiener process (Brownian motion)
The Bessel process of order n is the real-valued process X given (when n âÂÂ¥ 2) by
where ||÷|| denotes the Euclidean norm in R<sup>n</sup> and W is an n-dimensional Wiener process (Brownian motion). Note that this SDE makes sense for any real parameter (although the drift term is singular at zero).
A notation for the Bessel process of dimension started at zero is .
For n âÂÂ¥ 2, the n-dimensional Wiener process started at the origin is transient from its starting point: with probability one, i.e., X<sub>t</sub> > 0 for all t > 0. It is, however, neighbourhood-recurrent for n = 2, meaning that with probability 1, for any r > 0, there are arbitrarily large t with X<sub>t</sub> < r; on the other hand, it is truly transient for n > 2, meaning that X<sub>t</sub> âÂÂ¥ r for all t sufficiently large.
For n ⤠0, the Bessel process is usually started at points other than 0, since the drift to 0 is so strong that the process becomes stuck at 0 as soon as it hits 0.
0- and 2-dimensional Bessel processes are related to local times of Brownian motion via the RayâÂÂKnight theorems.
The law of a Brownian motion near x-extrema is the law of a 3-dimensional Bessel process (theorem of Tanaka).