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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 1106 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) 1108 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. 1110 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]. References 1. M.T. Barlow, J.W. Pitman, M. Yor: Une extension multidimensionnelle de la loi de l'arc sinus. Sern. Probas. XXIII. 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Yor: Une explication du théorème de Ciesielski-Taylor. To appear in Ann. I.H.P. 27 (2) (1991) 30. J. Bertoin: Complements on the Hilbert transform and the fractional derivatives of Brownian local times. J. Math. Kyoto 30 (4) (1990) 651-670 31. P.J. Fitzsimmons and R.K. Getoor: On the distribution of the Hilbert transform of the local time of a symmetric Levy process. To appear in Ann. Prob.