Krylov–Bogolyubov theorem

In mathematics, a Bessel process, named after Friedrich Bessel, is a type of stochastic process. The Bessel process of order n is the real-valued process X given by

$X_{t}=\|W_{t}\|,$

where ||·|| denotes the Euclidean norm in Rⁿ and W is an n-dimensional Wiener process (Brownian motion) started from the origin. The n-dimensional Bessel process is the solution to the stochastic differential equation

$dX_{t}=dZ_{t}+{\frac {n-1}{2}}{\frac {dt}{X_{t}}}$

where Z is a 1-dimensional Wiener process (Brownian motion). Note that this SDE makes sense for any real parameter $n$ (although the drift term is singular at zero). Since W was assumed to have started from the origin the initial condition is X₀ = 0.

For n ≥ 2, the n-dimensional Wiener process is transient from its starting point: with probability one, X_t > 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_t < r; on the other hand, it is truly transient for n > 2, meaning that X_t ≥ r for all t sufficiently large.

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

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

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• Williams D. (1979) Diffusions, Markov Processes and Martingales, Volume 1 : Foundations. Wiley. ISBN 0-471-99705-6.

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