Bessel process
In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process. Formal definition. The Bessel process of order "n" is the real-valued process "X" given (when "n" ≥ 2) by where ||·|| denotes the Euclidean norm in R and "W" is an "n"-dimensional Wiener process (Brownian motion). For any "n", the "n"-dimensional Bessel process is the solution to the stochastic differential equation (SDE) where "Z" is a 1-dimensional Wiener process (Brownian motion). Note that this SDE makes sense for any real parameter formula_3 (although the drift term is singular at zero). Notation . A notation for the Bessel process of dimension started at zero is . In specific dimensions. For "n" ≥ 2, the "n"-dimensional Wiener process started at the origin is transient from its starting point: with probability one, i.e., "X" > 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" < "r"; on the other hand, it is truly transient for "n" > 2, meaning that "X" ≥ "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. Relationship with Brownian motion. 0- and 2-dimensional Bessel processes are related to local times of Brownian motion via the Ray–Knight theorems. The law of a Brownian motion near x-extrema is the law of a 3-dimensional Bessel process (theorem of Tanaka).