# Bessel processes

A family of continuous Markov processes (cf. Markov process) $( R _ {t} , t \geq 0 )$ taking values in $\mathbf R _ {+}$, parametrized by their dimension $\delta$.

When $\delta = d$ is an integer, $( R _ {t} ,t \geq 0 )$ may be represented as the Euclidean norm of Brownian motion in $\mathbf R ^ {d}$. Let $Q _ {x} ^ {d}$ be the law of the square, starting from $x \geq 0$, of such a process $( R _ {t} ,t \geq 0 )$, considered as a random variable taking values in $\Omega = C ( \mathbf R _ {+} , \mathbf R _ {+} )$. This law is infinitely divisible (cf. [a6] and Infinitely-divisible distribution). Hence, there exists a unique family $( Q _ {x} ^ \delta ;x \geq 0, \delta \geq 0 )$ of laws on $\Omega$ such that

$$\tag{a1 } Q _ {x} ^ \delta * Q _ {x ^ \prime } ^ {\delta ^ \prime } = Q _ {x + x ^ \prime } ^ {\delta + \delta ^ \prime } \textrm{ for all } \delta, \delta ^ \prime ,x,x ^ \prime \geq 0$$

( $*$ indicates the convolution of probabilities on $\Omega$), which coincides with the family $( Q _ {x} ^ {d} ,x \geq 0 )$, for integer dimensions $d$.

The process of coordinates $( X _ {t} ,t \geq 0 )$ on $\Omega$, under $Q _ {x} ^ \delta$, satisfies the equation

$$\tag{a2 } X _ {t} = x + 2 \int\limits _ { 0 } ^ { t } {\sqrt {X _ {s} } } {d \beta _ {s} } + \delta t, \quad t \geq 0,$$

with $( \beta _ {s} ,s \geq 0 )$ a one-dimensional Brownian motion. Equation (a2) admits a unique strong solution, with values in $\mathbf R _ {+}$. Call its square root a $\delta$- dimensional Bessel process.

Bessel processes also appear naturally in the Lamperti representation of the process $( { \mathop{\rm exp} } ( B _ {t} + \nu t ) ,t \geq 0 )$, where $\nu \in \mathbf R$ and $( B _ {t} ,t \geq 0 )$ denotes a one-dimensional Brownian motion. This representation is:

$$\tag{a3 } { \mathop{\rm exp} } ( B _ {t} + \nu t ) = R _ {\int\limits _ { 0 } ^ { t } { { \mathop{\rm exp} } ( 2 ( B _ {s} + \nu s ) ) } {ds } } , t \geq 0,$$

where $R$ is a $\delta = 2 ( 1 + \nu )$- dimensional Bessel process. This representation (a3) has a number of consequences, among which absolute continuity properties of the laws $Q _ {x} ^ \delta$ as $\delta$ varies and $x >0$ 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 $( S _ {t} - B _ {t} ,t \geq 0 )$, and by J. Pitman as $( 2S _ {t} - B _ {t} ,t \geq 0 )$, where $S _ {t} = \sup _ {s \leq t } B _ {s}$, and $( B _ {t} ,t \geq 0 )$ is a one-dimensional Brownian motion.

Finally, the laws of the local times of $( B _ {t} ,t \geq 0 )$ considered up to first hitting times, or inverse local times, can be expressed in terms of $Q _ {0} ^ {2}$ and $Q _ {x} ^ {0}$, 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.

#### References

 [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. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Bessel_processes&oldid=46038
This article was adapted from an original article by M. Yor (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article