Bessel process

The Bessel process of order n is the real-valued process X given (when n ≥ 2) by :X_t = \| W_t \|, where ||·|| denotes the Euclidean norm in Rn and W is an n-dimensional Wiener process (Brownian motion). Note that this SDE makes sense for any real parameter n (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., Xt > 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 Xt  2, meaning that Xt ≥ 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). ==References==