Bessel processes

A family of continuous Markov processes (cf. Markov process) taking values in , parametrized by their dimension .

When is an integer, may be represented as the Euclidean norm of Brownian motion in . Let be the law of the square, starting from , of such a process , considered as a random variable taking values in . This law is infinitely divisible (cf. [a6] and Infinitely-divisible distribution). Hence, there exists a unique family of laws on such that (a1)

( indicates the convolution of probabilities on ), which coincides with the family , for integer dimensions .

The process of coordinates on , under , satisfies the equation (a2)

with a one-dimensional Brownian motion. Equation (a2) admits a unique strong solution, with values in . Call its square root a -dimensional Bessel process.

Bessel processes also appear naturally in the Lamperti representation of the process , where and denotes a one-dimensional Brownian motion. This representation is: (a3)

where is a -dimensional Bessel process. This representation (a3) has a number of consequences, among which absolute continuity properties of the laws as varies and is fixed, and also the fact that a power of a Bessel process is another Bessel process, up to a time-change.

Special representations of Bessel processes of dimensions one and three, respectively, have been obtained by P. Lévy, as , and by J. Pitman as , where , and is a one-dimensional Brownian motion.

Finally, the laws of the local times of considered up to first hitting times, or inverse local times, can be expressed in terms of and , respectively: this is the content of the celebrated Ray–Knight theorems (1963; [a1], [a5]) on Brownian local times. These theorems have been extended to a large class of processes, including real-valued diffusions.