Bessel processes

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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


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

The process of coordinates on , under , satisfies the equation


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:


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.


[a1] F.B. Knight, "Random walks and a sojourn density process of Brownian motion" Trans. Amer. Math. Soc. , 107 (1963) pp. 56–86
[a2] J.W. Pitman, "One-dimensional Brownian motion and the three-dimensional Bessel process" Adv. Applied Probab. , 7 (1975) pp. 511–526
[a3] J.W. Pitman, M. Yor, "Bessel processes and infinitely divisible laws" D. Williams (ed.) , Stochastic Integrals , Lecture Notes in Mathematics , 851 , Springer (1981)
[a4] J.W. Pitman, M. Yor, "A decomposition of Bessel bridges" Z. Wahrscheinlichkeitsth. verw. Gebiete , 59 (1982) pp. 425–457
[a5] D.B. Ray, "Sojourn times of a diffusion process" Ill. J. Math. , 7 (1963) pp. 615–630
[a6] T. Shiga, S. Watanabe, "Bessel diffusions as a one-parameter family of one-dimensional diffusion processes" Z. Wahrscheinlichkeitsth. verw. Gebiete , 27 (1973) pp. 37–46
[a7] D. Revuz, M. Yor, "Continuous martingales and Brownian motion" , Springer (1994) (Edition: Second)
How to Cite This Entry:
Bessel processes. M. Yor (originator), Encyclopedia of Mathematics. URL:
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098