Bessel process

For any n, the n-dimensional Bessel process is the solution to the stochastic differential equation (SDE) where W is a 1-dimensional Wiener process (Brownian motion).

Note that this SDE makes sense for any real parameter

A notation for the Bessel process of dimension n started at zero is BES0(n).

It is, however, neighbourhood-recurrent for n = 2, meaning that with probability 1, for any r > 0, there are arbitrarily large t with Xt < r; on the other hand, it is truly transient for n > 2, meaning that Xt ≥ r for all t sufficiently large.

0- and 2-dimensional Bessel processes are related to local times of Brownian motion via the Ray–Knight theorems.

Three realizations of Bessel Processes.