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A family of continuous Markov processes (cf. [[Markov process|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|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|random variable]] taking values in  $  \Omega = C ( \mathbf R _ {+} , \mathbf R _ {+} ) $. 
This law is infinitely divisible (cf. [[#References|[a6]]] and [[Infinitely-divisible distribution|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; [[#References|[a1]]], [[#References|[a5]]]) on Brownian local times. These theorems have been extended to a large class of processes, including real-valued diffusions.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  F.B. Knight,   "Random walks and a sojourn density process of Brownian motion"  ''Trans. Amer. Math. Soc.'' , '''107'''  (1963)  pp. 56–86</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J.W. Pitman,   "One-dimensional Brownian motion and the three-dimensional Bessel process"  ''Adv. Applied Probab.'' , '''7'''  (1975)  pp. 511–526</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.W. Pitman,   M. Yor,   "Bessel processes and infinitely divisible laws"  D. Williams (ed.) , ''Stochastic Integrals'' , ''Lecture Notes in Mathematics'' , '''851''' , Springer  (1981)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  J.W. Pitman,   M. Yor,   "A decomposition of Bessel bridges"  ''Z. Wahrscheinlichkeitsth. verw. Gebiete'' , '''59'''  (1982)  pp. 425–457</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  D.B. Ray,   "Sojourn times of a diffusion process"  ''Ill. J. Math.'' , '''7'''  (1963)  pp. 615–630</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  D. Revuz,   M. Yor,   "Continuous martingales and Brownian motion" , Springer  (1994)  (Edition: Second)</TD></TR></table>

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