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The Laws of Some Brownian Functionals
Marc Yor
Laboratoire de Probabilités, Université P. et M. Curie, Tour 56, 3ème étage
4, place Jussieu, F-75252 Paris Cedex 05, France
Thanks mainly to the relationship between the heat equation, newtonian potential
theory and Brownian motion, the laws of a large number of Brownian functionals
have been obtained during the last fifty years, at least via explicit expressions of
their Laplace and Fourier transforms. Much pioneering work in this area was done
by Paul Levy.
Gradually, with the development of Itô's stochastic calculus, excursion theory,
path decompositions and the technique of enlargement of nitrations, these studies
of individual distributions on IR, sometimes exhibiting identities between two laws,
which looked a priori to be mere "coincidences", have been understood in a deeper
way, in fact often by showing that two processes are identical in law; see Biane [3],
for a recent survey in that spirit.
The most elementary examples of Brownian functionals are linear functionals:
iff e L2(JR+, dt), and (Bt, t > 0) is a real-valued BM, the Wiener integral $ f(t) dBt
is a centered Gaussian variable, with variance J o / 2 ( 0 dt. Quadratic functionals of
BM represent the next level of complexity; those functionals are of great interest as,
somewhat surprisingly, they occur in a number of very different studies of Brownian
motion, such as the Ray-Knight theorems for Brownian local times, the CiesielskiTaylor identities, some limiting laws of planar BM, and principal values of Brownian
local times.
We shall take here, as a prototype of a quadratic Brownian functional, the
stochastic area of planar BM, and it will be shown how Paul Levy's formula for
this stochastic area appears again and again in most of the above mentioned studies
of Brownian motion.
1. On Levy's Area Formula
Consider a two-dimensional Brownian motion Z, = Xt + iYt91 > 0, starting from
0, and the stochastic area process
-i:
(X.dY.-Y.dX,),
t;>0.
Proceedings of the International Congress
of Mathematicians, Kyoto, Japan, 1990
© The Mathematical Society of Japan, 1991
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MarcYor
Levy's formula:
ECexpfttSJIZi =z] = \-^j)
exp - ^-(X coth X-l)
(1)
has played some important rôle in recent years, for example in the Bismut approach
[7, 8] to the Atiyah-Singer theorems.
To prove formula (1), Levy [14] used a diagonalization procedure. A different
approach (Williams [25], Yor [28]) is to use a change of probability method, which
reduces the computation of the law of the quadratic functional Sl to that of the
variance of a Gaussian variable.
First, by the rotational invariance of the law of BM, and independence properties, we have, for any X e IR:
Elexp^XS^Z,
=z] = EUxp - y J
ds\Zs\2\\Zx\ =
(2)
ds\Zs\2}-P\^
(3)
Next, we introduce the new probability P A :
^ U = expH(|Z ( | 2 -20-y
0
under which (Zt, t < 1) is a Gaussian process (more precisely: an Ornstein-Uhlenbeck
process) for which the variance at time 1 is easily computed. Formula (1) now follows
from formulae (2) and (3).
We note a simple consequence of formulae (1) and (2):
E exp-[a\Z1\2+
X2 •i
—
o
ds|Z, 1 2
.
, „
sinh X\ *
= cosh A + 2 a — —
.
...
(4)
Many variants of Levy's formula (1) have now been developed; in particular,
Biane-Yor [5] obtained a sequence of extensions of Levy's formula by decomposing
(Zu, u < 1) into the sum of the Brownian bridge (Zu — uZu u <1) and (uZu u < 1),
then developing the left-hand side of (1) with respect to this decomposition, and
finally iterating this procedure. The Levy formulae obtained in this way are closely
connected, on the one hand, with the linear decomposition of BM along (the
orthogonal basis of the Legendre polynomials and, on the other hand, with the
continued fraction:
,
, , ,
X2 X2 X2
X coth X — 1 = -— -— -— • • •.
3+ 5 + 7 +
2. Squares of Bessel Processes and Ray-Knight Theorems
Let ò > 1 be an integer, and (qt, t > 0) be the square of a BES(<5) process, that is the
square of the euclidean norm of a <5-dimensional BM (Bt, t > 0); then, q satisfies the
The Laws of Some Brownian Functionals
1107
SDE:
4, = <7o + 2
Jqsdßs + Öt,
(5)
Jo
where (ßs, s > 0) is a real-valued BM.
It is well-known that this equation, with q0 = x > 0, and ö any positive real
number, has a unique pathwise solution in R + , hence a unique law on C(R+, R + ),
which we shall denote by Qx.
Shiga and Watanabe [23] remarked that
Qi * Gî' = GÎÎÎ', for any 5, 3', x, x' > 0
(6)
(where P * ß is the convolution of P and Q), thus extending to all starting points
and dimensions the obvious additivity property for integer dimensions.
From (6), Qx is an infinitely divisible probability distribution on C(R + , R + ),
which admits the following Lévy-Khintchine representation (Pitman-Yor [18]):
there exist two a-finite > 0 measures M and JV on C(R + , R + ) such that:
ßi(exp - (co, / » = exp - (xM + ÖN)(1 - exp - <co, / »
(7)
(we use the notation: (co, / > = j£ dt co(t)f(t), for / > 0).
This representation may be obtained, including an explicit description of M and
N in terms of the Ito characteristic measure n(dco) of the Poisson point process of
Brownian excursions, with the help of the Ray-Knight theorems on Brownian local
times (Lat\ a e R, t > 0), which are now presented:
(RKJ
if Ti = inf{t : Bt = 1}, the law of (L 1 ^; 0 < a < 1) is Q20
(RK 2 )
if TX = inf{t : L? - x}9 the law of (L\x\ h > 0) is Q°x
(after the original proofs of Ray [22] and Knight [12], several different derivations
of (RKJ and (RK 2 ) have been given; see, for example, Jeulin [11] for a recent survey
based on Tanaka's formula).
Conversely, we may now deduce from the Lévy-Khintchine representation (7)
some extensions of the Ray-Knight theorems; here is one (Le Gall-Yor [13]): for
ô > 0, the law of the process (C^}; a > 0) of the local times of ( \Bt\ + -L?; t > 0
is ßg-
^
6
In the particular case ö = 2, we recover (RKJ, using the representation of
BES(3) as (\Bt\ + L?; t > 0), due to Pitman ([17]), jointly with a well-known timereversal relation between BES(3) and Brownian motion (see Williams [26]).
3. An Explanation of the Ciesielski-Taylor Identities
Ciesielski-Taylor [9] obtained the following puzzling identity in law:
1
dsW^i)°=TM,
0
(8)
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MarcYor
where Rô resp. Rô+2, is a BES process, with dimension <5, resp. 5 + 2, starting from
0,andTl(Rô) =
inf{f.R0(t)=l}.
In the case ö = 1, the identity (8) can be understood by time-reversal, but
no such pathwise explanation has been obtained for other dimensions. Below, a
spectral-type explanation and extensions are presented, following [29].
Writing both sides of (8) as integrals of the local times of the two BES processes,
and using the Ray-Knight theorems, (8) then appears as a particular case of the
identity in law:
- f ' daf\a)B2m **> \1 da g'(a)B}(a)
(9)
Jo
Jo
where f,g : [0, 1] -> R + , are C 1 , with /decreasing, # increasing, and f(\) = g(0) = 0.
In fact, a more general identity in law holds:
dxf(x)B2g{x) + f(h)B2m
{la ]
= g(a)B2{a) + j & dx g'(x)B2(x)
(10)
where /, g : [a, b] -> R + , are C 1 , with / decreasing, and g increasing. In turn, this
implies some extensions of the C-T identities.
Now, it is easily shown that (10) is a particular case of the following Fubini-type
identity in law:
foo
/ f oo
dsl\
\2
dBucp(s,u)) (^w)
f oo
dBMu,s))2
ds
(11)
0
where <p e L 2 ([0, oo[ 2 ;R)
A striking application of (11) is the following identity in law, obtained with C.
Donati-Martin:
J:
ds(B s -G) 2 ( = w )
dsM-sBtf,
(11')
0
where G = JJdw Bu.
The general identity (11) is an infinite dimensional extension of the elementary
identity in law: if Xn = (Xl9..., Xn) is an n-dimensional sample of the N(0, a2) law,
then, for any n x n matrix:
Hitìfi, || (1 = ) \\A*Xn\\,
where A* is the transpose of A.
(12)
The above arguments may be developed to give an explanation of the large class
of extensions of the C-T identities obtained by Biane [4], between functionals of
pairs of diffusions which satisfy a certain duality property.
4. Some Limiting Laws of Planar BM
Let (Zt, t > 0) be a planar BM starting from z 0 , and consider (6t, t > 0) a continuous
determination of the argument of (Zu, u < t) around zx ^ z0.
The Laws of Some Brownian Functionals
1109
Spitzer [24] showed that:
2
ft-g^Q
log*
(13)
where Cx is a standard Cauchy variable.
This may be refined by decomposing 0, into: 0f~ + 0+, where:
0~ =
d0.1<|z.-*,|<;in
and
ö +
r =
^W^l^)
1
Jo
Jo
Then, considering moreover (At, t > 0) an integrable additive functional, we have
(Messulam-Yor [16]):
2
log
-(e-,et\At)-^(w-,w+,cAA)
where: W~ = fgrfy.Wo), ^ + = j o ^ W o ) * ^ = '°> <U i s a constant depending
only on A, ß and y are two independent linear BM's starting from 0, cr = inf {t : ßt = 1},
and (/°, u ^ 0) is the local time of ß at 0. The law of (W~, W+, A) is characterized by:
£[exp(-flyl -f ibW~ + icW*)'] = (cosh c + J L L U s i n n
c
j
(14)
a formula which is very similar to (4), this being easily explained thanks mainly to
the Ray-Knight theorem (RKX).
The convergence in law (13) may be further extended by considering ( 0 / , . . . , 6";
t > 0), the winding numbers of (ZM, u < t) around n distinct points zl9..., zn.
One obtains (Pitman-Yor [19, 20]):
^,...,en^^(w1,...,wn)
and the characteristic function of (Wl9..., Wn) is:
E exp t(j£
XjW^J = ( c o s h ( è X^j + Ç ^ l s i n h ( t X^j
\
(15)
5. Arc sine Laws for Linear Brownian Motion
Let (Bt, t > 0) be the linear BM, starting from 0. Levy [15] showed that: F + =
J J ds 1(BS>O) follows the arc sine distribution, that is:
P(r+ed0 = - ^ 4 = v
( 16 >
7iy/t(i - o
(Notice that, in paragraph 4, we considered \a0ds 1(BSZO)> where a = inf{t : Bt = 1};
now, G is replaced by 1).
The r.v. gx = sup{t < 1 : Bt = 0} is also arc sine distributed, but this is easier
to prove.
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MarcYor
Here is a recent proof of (16), obtained with M. Barlow and J. Pitman [1], using
excursion theory; one can show:
l<r.(t\
r w (hlw)
^ ( A « , 71(0)
= (T+9T-\
(17)
where T+ and T_ are two independent stable (^) variables.
It now follows that:
r+ = r+{l)^jr^>
which proves (16).
Several infinite dimensional extensions of (17) have now been obtained, jointly
with J. Pitman [21]. Here is one: let V(t) be the infinite sequence of lengths of
excursions of B away from 0, during the time interval (0, t), including the last
unfinished excursion, arranged in decreasing order, so that:
v(t) = (v,(t),...,
VM.--1
with vx(t)> v2(t)>
> vn(t)> •••.
Then, for every t > 0, and s > 0,
where (TS, S > 0) is the inverse of the local time (lt,t> 0).
6. Cauchy's Principal Value of Brownian Local Times
Consider again (Bt, t > 0) a linear BM starting from 0, (lt, t > 0) its local time at 0,
and (xt, t > 0) the inverse of /. As a consequence of the regularity of Brownian local
times,
l
ds
i* t d =lim
e->0
m
0
-^\\B
S\>&)
D
s
exists a.s., uniformly in t i n a compact set I whereas:
\
ds
- ^ - = oo a.s. ). From the
JO \Bs\
scaling property and the Markov property of BM, it follows that: (HZt, t > 0) is a
symmetric Cauchy process (with parameter n\).
Independently, it was remarked by Spitzer [24] that if (Xu, u > 0) is a linear BM
independent of (nt, t > 0), then (Xtt, t > 0) is a symmetric Cauchy process.
However, this identity in law does not extend to the two 2-dimensional processes:
-HXt, Tt; t > 0J
and
(XXt, zt; t > 0)
since we have the formula:
EUxM^HZt-B-xA
From this formula, we deduce:
=expf-acothQjJ.
(19)
The Laws of Some Brownian Functionals
E exp( i-HT }\lT = x
1111
X
sinh X exp - x(X coth X - 1)
(20)
where T is an exponential variable (with parameter (^)) independent of B. From
Levy's formula (1), we deduce:
1 „
, . ( l a w ) / « . l.rj
,2
^/f^W^MS^-IZJ2)
(21)
which has not yet received a simple direct explanation.
As a consequence, we obtain, for fixed t > 0:
2 \ 1 / 2 Ä , ...
/ /
P(HtedX) = [^t)
Z(-ir^-{n
lVx'
+ -) ~)d,
(22)
The fact that some relation between the laws of the processes (Ht,t > 0) and
(St,t> 0) exists may be understood, at least in some sense, via easier identities in
law, such as:
dS(law) 9
R
"
•1
\-l/2
2
QdsR
)
,
(23)
where (Rt,t> 0), resp: (Rt, t > 0) is a BES process with dimension ö > 2, resp:
S = 20 - 2.
All the results presented in this paragraph are taken from Biane-Yor [6].
Principal values of Brownian local times have been studied in depth by Yamada
(see, in particular, [27]) and Bertoin [2]; they have also been investigated, for
physical purposes, by Ezawa et al [10].
Note added in proof: Further results about Cauchy's principal value of local times
have now been obtained by Bertoin [30] and Fitzsimmons and Getoor [31].
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