Celebratio Mathematica

When I first met Marc Yor around 1977, the the­ory of ex­cur­sions of Browni­an mo­tion was at an early stage of de­vel­op­ment, with few ap­plic­a­tions. The the­ory had been ini­ti­ated by Lévy [◊], and de­veloped fur­ther by Itô-McK­ean [◊], Sec­tion 2.9. Dav­id Wil­li­ams [◊] made the key ob­ser­va­tion that a Browni­an ex­cur­sion of length \( t \) is identic­al in law to a \( \operatorname{BES}(3) \) bridge from 0 to 0 of length \( t \). Here \( \operatorname{BES}(\delta) \) for \( \delta = 1,2, \ldots \) stands for a \( \delta \)-di­men­sion­al Bessel pro­cess, mean­ing the ra­di­al part of a \( \delta \)-di­men­sion­al Browni­an mo­tion. Wil­li­ams [◊] in­tro­duced the key idea of de­com­pos­ing the path of a one-di­men­sion­al dif­fu­sion at the time of its max­im­um. This iden­ti­fied vari­ous path frag­ments in Browni­an mo­tion with oth­er path frag­ments de­rived from a \( \operatorname{BES}(3) \) pro­cess. \( \operatorname{BES}(3) \) then came to be un­der­stood as a con­veni­ent rep­res­ent­a­tion of Browni­an mo­tion on the pos­it­ive half-line con­di­tioned nev­er to re­turn to 0, in a sense made pre­cise by Doob’s the­ory of \( h \)-trans­forms. Then Chung [◊] provided a de­tailed de­vel­op­ment of Lévy’s the­ory of ex­cur­sions strad­dling a fixed time, em­phas­ising par­al­lels with the the­ory of last exit de­com­pos­i­tions for Markov chains. But it was not un­til some years later that Wil­li­ams [◊] ex­plained the im­port­ance of his path de­com­pos­i­tions for the the­ory of Browni­an ex­cur­sions. Around this time, I was in­ter­ested in the the­ory of path de­com­pos­i­tions and ex­cur­sions, in the gen­er­al Markovi­an set­ting of Itô [◊] and Mais­on­neuve [◊], while Marc was en­gaged with Jacques Azéma in work on loc­al times of semi-mar­tin­gales, the balay­age for­mula as­so­ci­ated with last exit times, and the Skorok­hod em­bed­ding prob­lem. Marc and I were both deeply im­pressed by the work of Dav­id Wil­li­ams. We sensed there was much to be done to flesh out Wil­li­ams’ ideas, which re­lated vari­ous path de­com­pos­i­tions of Browni­an mo­tion to the Ray–Knight rep­res­ent­a­tions of Browni­an loc­al time pro­cesses as squares of Bessel pro­cesses. This shared in­terest in the Browni­an world was the start­ing point of a col­lab­or­a­tion which con­tin­ued for over 25 years. This col­lab­or­a­tion also in­volved nu­mer­ous oth­er stu­dents and re­search­ers, both in Berke­ley, where Marc vis­ited reg­u­larly each sum­mer, and in Par­is where I spent a num­ber of sab­bat­ic­als.

In the early years of our col­lab­or­a­tion, I provided ex­pert­ise in Markovi­an ex­cur­sion the­ory, while Marc was the mas­ter of mar­tin­gale cal­cu­lus. There was some friendly com­pet­i­tion across the At­lantic, wheth­er the laws of Browni­an func­tion­als were best com­puted by stochast­ic cal­cu­lus or ex­cur­sion the­ory. But it was not long be­fore Marc was the mas­ter and I was the ap­pren­tice in both frame­works.

The per­spect­ives of ex­cur­sion the­ory and of mar­tin­gale cal­cu­lus are very dif­fer­ent. In ex­cur­sion the­ory, the Browni­an path is viewed as a con­cat­en­a­tion of a count­able col­lec­tion of ran­dom path frag­ments, each with its own in­tern­al dy­nam­ics, with tim­ing of the con­cat­en­a­tion by a loc­al time clock. The frag­ments are dot­ted around some space of ex­cur­sion paths, in which they can read­ily be sub­jec­ted to vari­ous op­er­a­tions such as time-re­versal, time-changes or oth­er trans­form­a­tions. By com­par­is­on, to ap­ply mar­tin­gale cal­cu­lus, you start with a giv­en fil­tra­tion, work with pro­cesses ad­ap­ted to that fil­tra­tion, en­lar­ging the fil­tra­tion when ne­ces­sary to ac­com­mod­ate ran­dom times that are not stop­ping times. It is not so easy in this frame­work to cut up pieces of path and re­as­semble them, and have any idea what pro­cess has been cre­ated. Still, as Marc showed over and over again, each of these frame­works has much to of­fer the oth­er. The key to many of Marc’s deep­est con­tri­bu­tions to the the­ory of Browni­an mo­tion and re­lated pro­cesses was his mas­tery of both mar­tin­gale cal­cu­lus and ex­cur­sion the­ory.

This art­icle of­fers some hig­lights of Marc Yor’s work in the the­ory of ex­cur­sions of Browni­an mo­tion and re­lated pro­cesses, es­pe­cially Bessel pro­cesses. See the art­icles of Ber­toin [◊] and Le Gall [◊] for brief in­tro­duc­tions to the con­text of this work. I fol­low quite closely the present­a­tions of Pit­man–Yor [◊] and [◊]. For much more ex­tens­ive ac­counts of this work, see these art­icles and the mono­graphs Re­vuz–Yor [◊], Yor [◊], Yen–Yor [◊], and Mal­lein–Yor [◊].

In his fun­da­ment­al pa­per [◊], Itô showed how to con­struct a Pois­son point pro­cess of ex­cur­sions of a strong Markov pro­cess \( X \) over time in­ter­vals when \( X \) is away from a re­cur­rent point \( a \) of its statespace. The point pro­cess is para­met­er­ized by the loc­al time pro­cess of \( X \) at \( a \). Each point of the ex­cur­sion pro­cess is a path in a suit­able space of pos­sible ex­cur­sions of \( X \), start­ing at \( a \) at time 0, and re­turn­ing to \( a \) for the first time at some strictly pos­it­ive time \( \zeta \), called the life­time of the ex­cur­sion. The in­tens­ity meas­ure of the Pois­son pro­cess of ex­cur­sions is a \( \sigma \)-fi­nite meas­ure \( \Lambda \) on the space of ex­cur­sions, known as Itô’s ex­cur­sion law. For a re­flect­ing Browni­an mo­tion \( X \) on \( [0,\infty) \), there are the fol­low­ing two fun­da­ment­al de­scrip­tions of Itô’s law \( \Lambda \) for ex­cur­sions away from 0. The first de­scrip­tion is drawn from Itô’s defin­i­tion and ob­ser­va­tions of Lévy [◊], Itô-McK­ean [◊], and Wil­li­ams [◊]. The second de­scrip­tion is due to Wil­li­ams [◊] and proved in Ro­gers [◊].

De­scrip­tion I: Con­di­tion­ing on the life­time:
First pick a life­time \( t \) ac­cord­ing to the \( \sigma \)-fi­nite dens­ity \( ({2 \pi})^{-1/2} t^{-3/2} \,dt \) on \( (0, \infty) \); then giv­en \( t \), run a \( \operatorname{BES}(3) \) bridge from 0 to 0 over time \( t \).

De­scrip­tion II: Con­di­tion­ing on the max­im­um:
First pick a max­im­um value \( m \) ac­cord­ing to the \( \sigma \)-fi­nite dens­ity \( m^{-2} \,dm \) on \( (0, \infty) \); then giv­en \( m \), join back to back two in­de­pend­ent \( \operatorname{BES}(3) \) pro­cesses, each star­ted at 0 and run till it first hits \( m \).

Pit­man–Yor [◊] gen­er­al­ized the equi­val­ence of these two de­scrip­tions of Itô’s law of pos­it­ive Browni­an ex­cur­sions as fol­lows. The equi­val­ence in­volving \( \operatorname{BES}(3) \) is ex­ten­ded to one in­volving \( \operatorname{BES}(\delta) \), as defined by Shiga–Watanabe [◊] for all real \( \delta \ge 0 \) us­ing ad­dit­iv­ity prop­er­ties of squares of Bessel pro­cesses, and as dis­cussed fur­ther in Le Gall [◊], Sec­tion 5.

The meas­ures \( \Lambda_{00}^\delta \) defined by De­scrip­tion II for \( \delta > 2 \) were con­sidered already by Pit­man–Yor [◊] and fur­ther stud­ied by Bi­ane–Yor [◊], who gave De­scrip­tion I in this case. It was shown in Pit­man–Yor [◊] that for \( 2 < \delta < 4 \) the meas­ure \( \Lambda_{00}^\delta \) is Itô’s ex­cur­sion law for ex­cur­sions of \( \operatorname{BES}(4-\delta) \) away from zero, up to mul­ti­plic­a­tion by a con­stant de­pend­ing on the nor­mal­iz­a­tion of loc­al time. In par­tic­u­lar, if \( n = n_{+} + n_{-} \) is the de­com­pos­i­tion of Itô’s ex­cur­sion law for ex­cur­sions of stand­ard Browni­an mo­tion in­to its parts for pos­it­ive and neg­at­ive ex­cur­sions, with the usu­al nor­mal­iz­a­tion of loc­al time as oc­cu­pa­tion dens­ity re­l­at­ive to Le­besgue meas­ure, as in Re­vuz–Yor [◊], Sec­tion XII.4, then \[ \Lambda_{00}^3 = 2 n_{+} \] and The­or­em 1 re­duces to the pre­ced­ing de­scrip­tions of Itô’s law \( \Lambda = 2 n_{+} \) for ex­cur­sions of a re­flect­ing Browni­an mo­tion.

For all \( \delta \ge 2 \) the meas­ure \( \Lambda_{00}^\delta \) con­cen­trates on ex­cur­sion paths start­ing at 0 and first re­turn­ing to 0 at their life­time. But the meas­ure with dens­ity \( t^{-\delta/2} \,dt \) on \( (0, \infty) \) is a Lévy meas­ure only for \( 2 < \delta < 4 \). So for \( \delta \le 2 \) or \( \delta \ge 4 \) the meas­ure \( \Lambda_{00}^\delta \) is not the ex­cur­sion law of any Markov pro­cess. Non­ethe­less, these meas­ures \( \Lambda_{00}^\delta \) are well defined for all \( \delta > 0 \), and have some in­ter­est­ing prop­er­ties and ap­plic­a­tions. It was shown in in Pit­man–Yor [◊] that the meas­ure \( N \) defined as the dis­tri­bu­tion of the square of the path un­der \( 2 \Lambda_{00}^4 \) ap­pears, due to the Ray–Knight de­scrip­tion of Browni­an loc­al times, as the dis­tri­bu­tion of the total loc­al time pro­cess of a path gov­erned by the Itô’s law for pos­it­ive Browni­an ex­cur­sions \( n_+ = \frac{1}{2} \Lambda_{00}^3 \). As a con­sequence, the same meas­ure \( N \) ap­pears in the Lévy–Kh­intchine rep­res­ent­a­tion of the in­fin­itely di­vis­ible fam­ily of squares of Bessel pro­cesses and Bessel bridges found in Pit­man–Yor [◊], and de­scribed in Le Gall [◊], Sec­tion 5.

For \( 0 < \delta < 2 \), the point 0 is a re­cur­rent point for \( \operatorname{BES}(\delta) \), and the meas­ure \( \Lambda_{00}^\delta \) con­cen­trates on paths which, un­like ex­cur­sions, re­turn many times to 0 be­fore fi­nally be­ing killed at 0. This \( \sigma \)-fi­nite meas­ure can be de­scribed in an­oth­er way as fol­lows: for a suit­able nor­mal­iz­a­tion of loc­al time, the total loc­al time at 0 is dis­trib­uted ac­cord­ing to Le­besgue meas­ure, and giv­en that this loc­al time is \( \ell \) the path is dis­trib­uted like \( \operatorname{BES}(\delta) \) star­ted at 0 and run un­til in­verse loc­al time \( \ell \). Such meas­ures were first con­sidered in Pit­man–Yor [◊], Re­mark (3.9), and they played a role in the work of Bi­ane–Yor [◊] and Bi­ane–Le Gall–Yor [◊]. Closely re­lated \( \sigma \)-fi­nite meas­ures on paths with in­fin­ite life­time ap­pear also in the study of lim­it laws ob­tained for Browni­an pen­al­isa­tions by Na­j­nudel–Roynette–Yor [◊], [◊]. See also the re­cent work by Marc’s stu­dents Na­j­nudel–Nik­egh­bali [◊] for gen­er­al­iz­a­tions and vari­ations of this theme, whereby for a suit­able non-neg­at­ive sub­martin­gale \( (X_t) \) re­l­at­ive to a filtered prob­ab­il­ity space \( (\Omega, \mathcal{F}, P, (\mathcal{F}_t)_{t \ge 0}) \) sat­is­fy­ing some tech­nic­al con­di­tions, there is a \( \sigma \)-fi­nite meas­ure \( Q \) on \( (\Omega, \mathcal{F}) \) such that for all \( t \ge 0 \) and all events \( F_t \in \mathcal{F}_t \), \[ Q( F_t, g \le t ) = P ( 1_{F_t} X_t ) \] where \( g \) is the last time that \( X \) hits 0.

In Pit­man–Yor [◊] we es­tab­lished The­or­em 1 for all \( \delta > 0 \) us­ing a gen­er­al for­mu­la­tion of Wil­li­ams’ path de­com­pos­i­tion at the max­im­um for one-di­men­sion­al dif­fu­sion bridges, due to Fitz­sim­mons [◊]. As an ap­plic­a­tion of this de­com­pos­i­tion, fol­low­ing the trail blazed for \( \delta =3 \) by Bi­ane–Yor [◊], we de­scribed the law of the stand­ard \( \operatorname{BES}(\delta) \) bridge by its dens­ity on path space re­l­at­ive to the law ob­tained by tak­ing two in­de­pend­ent \( \operatorname{BES}(\delta) \) pro­cesses star­ted at 0 and run till they first hit 1, join­ing these pro­cesses back to back, and scal­ing the res­ult­ant pro­cess with a ran­dom life­time and max­im­um 1 to have life­time 1 and a ran­dom max­im­um.

Note es­pe­cially the most im­port­ant in­stance of the agree­ment for­mula \eqref{ran.5}, which arises with \( F(R^{\mathrm{br}} ) \) and \( F( \tilde{R}^{\mathrm{br}} ) \) in \eqref{ran.5} re­placed by \( F(M^{\mathrm{br}} ) \) and \( F( \tilde{M}^{\mathrm{br}} ) \) where \( M^{\mathrm{br}} \) is the max­im­um of \( R^{\mathrm{br}} \), and the do­main of \( F \) is now \( [0,\infty) \) in­stead of \( C[0,1] \). This ab­so­lute con­tinu­ity re­la­tion between the laws of the \( M^{\mathrm{br}} \) and \( \tilde{M}^{\mathrm{br}} \) is the heart of the mat­ter, which is de­rived from the agree­ment for­mula for the \( \sigma \)-fi­nite meas­ures by a Fu­bini ar­gu­ment. This style of ar­gu­ment and the scope of the agree­ment for­mula were gradu­ally ex­ten­ded, start­ing from their first ap­pear­ances in work of Bi­ane–Le Gall–Yor [◊] and Bi­ane–Yor [◊]. See Pit­man–Yor [◊], [◊] for vari­ations of this spli­cing con­struc­tion for Bessel pro­cesses, such as re­pla­cing first pas­sage times by last pas­sage times, and for ex­pli­cit de­scrip­tions by in­fin­ite series of the dis­tri­bu­tion of the max­im­um of a Bessel bridge for all real \( \delta > 0 \).

Riemann [◊] showed that his zeta func­tion, ini­tially defined by the series \begin{equation} \label{zeta1} \zeta(s) :=\sum_{n=1}^{\infty}n^{-s} \quad (\Re s > 1) \end{equation} ad­mits a mero­morph­ic con­tinu­ation to the en­tire com­plex plane, with only a simple pole at 1, and that the func­tion \begin{equation} \label{xifn} \xi(s):= \textstyle\frac{1}{2} s ( s-1) \pi^{-s/2} \Gamma\bigl( \frac{1}{2} s \bigr) \zeta( s) \quad (\Re s > 1) \end{equation} is the re­stric­tion to \( (\Re s > 1) \) of a unique en­tire ana­lyt­ic func­tion \( \xi \), which sat­is­fies the func­tion­al equa­tion \begin{equation} \label{xieq} \xi(s)= \xi (1-s) \end{equation} for all com­plex \( s \). Riemann showed that these ba­sic prop­er­ties of \( \zeta \) and \( \xi \) fol­low from a rep­res­ent­a­tion of \( 2 \xi \) as the Mel­lin trans­form of a func­tion in­volving de­riv­at­ives of Jac­obi’s theta func­tion. Bi­ane–Yor [◊] iden­ti­fied this func­tion with the prob­ab­il­ity dens­ity of the dis­tri­bu­tion of the max­im­um of a Browni­an ex­cur­sion, first found by Chung and Kennedy, to con­clude that if \( M \) de­notes the max­im­um of a Browni­an ex­cur­sion of length 1, and \( Y:= \sqrt{2/\pi}M \), then \begin{equation} \label{y1} E ( Y^s ) = 2 \xi (s ) \quad (s \in \mathbb{C} ) \end{equation} Bi­ane–Yor showed that Riemann’s func­tion­al equa­tion \eqref{xieq} is a con­sequence of the agree­ment for­mula \eqref{ran.5} for the max­im­um for \( \delta=3 \), com­bined with Chung’s re­mark­able iden­tity in dis­tri­bu­tion that \begin{equation} \label{y2} Y^2 \stackrel{d}{=} \frac{\pi}{2} ( T + \hat{T} ) \end{equation} for \( T \) and \( \hat{T} \) the re­spect­ive hit­ting times of 1 of two in­de­pend­ent in­de­pend­ent \( \operatorname{BES}(3) \) pro­cesses start­ing at 0, as in Co­rol­lary 2 for \( \delta = 3 \). See Yen–Yor [◊], Sec­tion 11.6, for a de­riv­a­tion of Chung’s iden­tity via Browni­an ex­cur­sions, based on Yor [◊], Sec­tion 6.2. Many oth­er con­struc­tions of ran­dom vari­ables with the same dis­tri­bu­tion as \( Y \) have been dis­covered, in­volving func­tion­als of the path of a Browni­an mo­tion or Browni­an bridge in \( \mathbb{R}^d \) for \( d = 1,2,3 \) or 4. See Bi­ane–Pit­man–Yor [◊] for a sur­vey of such re­la­tions between the dis­tri­bu­tion of Browni­an func­tion­als and the Riemann zeta func­tion, and Pit­man–Yor [◊] for fur­ther de­vel­op­ments mo­tiv­ated by study of the dis­tri­bu­tion of ranked heights of ex­cur­sions of a Browni­an bridge.

For \( B \) a stand­ard Browni­an mo­tion star­ted at \( B_0 = 0 \), and \( t \ge 0 \), let \[ A_t := \int_0^t 1(B_t > 0 ) \,dt \] de­note the amount of time that \( B \) spends pos­it­ive up to time \( t \). It was no­ticed already by Lévy [◊] that dis­tri­bu­tion of \( A_T/T \) is the same, both for any fixed time \( T \), and for any in­verse loc­al time \( T = T_s := \inf \{t : L_t > s \} \), where \( L \) is the loc­al time pro­cess of \( B \) at 0. It is eas­ily shown, due to the in­de­pend­ence of pos­it­ive and neg­at­ive Browni­an ex­cur­sions, and Browni­an scal­ing, that dis­tri­bu­tion of \( A_{T_s}/T_s \) is identic­al to that of \( T_{s/2}/T_s \), where \( T_s = T_{s/2} + T_{s/2}^{\prime} \) for two in­de­pend­ent and identic­ally dis­trib­uted stable\( (1/2) \) vari­ables \( T_{s/2} \) and \( T_{s/2}^{\prime} \). It fol­lows that this com­mon dis­tri­bu­tion of \( A_{T_s}/T_s \) for all \( s > 0 \) is the arc­sine or beta\( (1/2,1/2) \) dis­tri­bu­tion, where beta\( (a,b) \) for \( a > 0, b > 0 \) is the prob­ab­il­ity dis­tri­bu­tion on \( [0,1] \) whose dens­ity at \( 0 < u < 1 \) is pro­por­tion­al to \( u^{a-1}(1-u)^{b-1} \). Lévy showed by sep­ar­ate com­pu­ta­tions that this arc­sine law was also the dis­tri­bu­tion of \( A_t/t \), as well as the dis­tri­bu­tion of \( G_t/t \), for each fixed time \( t \), where \( G_t:= \sup \{s \le t : B_s = 0 \} \), \( D_t:= \sup \{s > t : B_s = 0 \} \), so that \( (G_t, D_t) \) is the ex­cur­sion in­ter­val of \( B \) that straddles \( t \), and \( G_t < t < D_t \) with prob­ab­il­ity one for each fixed \( t \).

Why there should be an iden­tity in law between \( A_T/T \) between fixed times \( T \) and in­verse loc­al times \( T \) is not read­ily ap­par­ent. The point is that if \( T= T_s \) for some fixed \( s \), then \( B_T = 0 \) and \( G_T = T = D_T \) al­most surely, so there is no ex­cur­sion strad­dling \( T \). Con­sid­er­ing the time spent pos­it­ive \( A_T \) as a sum of times spent in pos­it­ive ex­cur­sions, for an in­verse loc­al time \( T \), the ex­cur­sions are all com­pleted. But for a fixed time \( T \) there is a col­lec­tion of com­pleted ex­cur­sions up to time \( G_T \), fol­lowed by an in­com­plete ex­cur­sion, known as a me­ander of length \( T- G_T > 0 \). There is no ob­vi­ous way to re­late the struc­ture of the par­ti­tion of \( [0,T] \) in­to ex­cur­sion in­ter­vals between fixed and in­verse loc­al times \( T \). So this iden­tity in dis­tri­bu­tion due to Lévy is not im­me­di­ately ex­plained by ex­cur­sion the­ory.

To provide an ad­equate ex­plan­a­tion of this iden­tity in terms of Browni­an ex­cur­sions, Pit­man–Yor [◊] con­sidered the se­quence \[ {\mathbf V}(t):=(V_1(t),V_2(t),\cdots), \quad V_1(t)\geq V_2(t)\geq\cdots, \] of ranked lengths of the max­im­al open subin­ter­vals of \( Z^c\cap(0,t) \), where \( Z \) is the clos­ure of the range of a sub­or­din­at­or \( (T_s, s \ge 0) \). In the stable case of in­dex \( 0 < \alpha < 1 \), we showed that \[ {\mathbf V}(t)/t \stackrel{d}{=} {\mathbf V}(T_s)/T_s \quad\text{for all} t > 0 \text{and} s > 0. \] In the case \( \alpha=\frac 12 \), when \( Z \) can be taken to be the zero set of a Browni­an mo­tion, Lévy’s iden­tity in dis­tri­bu­tion of \( A_T/T \) for fixed and in­verse loc­al times \( T \) is an im­me­di­ate con­sequence.

While for the range of a stable sub­or­din­at­or, the law of re­l­at­ive ranked lengths \( {\mathbf V}(T)/T \) is the same for both fixed and in­verse loc­al times \( T \), the way these in­ter­vals are ar­ranged is not the same in the two cases. For \( T= T_s \), the ar­range­ment forms an ex­change­able in­ter­val par­ti­tion of \( [0,T] \), mean­ing that (as­sum­ing for sim­pli­city \( V_1(T) > V_2(T) > \cdots \)) the longest in­ter­val of length \( V_1(T) \) is equally likely to be to the right of the left of the second longest in­ter­val of length \( V_2(T) \), the longest 3 in­ter­vals are equally in any of the \( 3! \) pos­sible or­ders, and so on. But for \( T \) a fixed time, there is the spe­cial me­ander in­ter­val \( (G_T,T] \), \( T- G_T = V_J(T) \) for some ran­dom in­dex \( J \). Re­mark­ably, for fixed times \( T \), the me­ander length \( V_J(T) \) turns out to be a sized biased pick from the se­quence \( {\mathbf V}(T) \), mean­ing that \[ P(J=j \,|\, {\mathbf V}(T) ) = V_j(T)/T \quad ( j = 1,2, \ldots) \] Once this spe­cial length is se­lec­ted from the ranked lengths to be placed on the ex­treme right, and even con­di­tion­ally on the length of this in­ter­val, the or­der of re­main­ing lengths, form­ing an in­ter­val par­ti­tion of \( [0,G_T] \), is that of an ex­change­able in­ter­val par­ti­tion. In the Browni­an case (\( \alpha = \frac{1}{2} \)) this is so be­cause the in­ter­val par­ti­tion of \( [0,G_T] \) is gen­er­ated by a Browni­an bridge of length \( G_T \), which con­di­tion­ally giv­en \( G_T \) has ex­change­able in­cre­ments. But the in­ter­val par­ti­ti­tion gen­er­ated by a stable sub­or­din­at­or has this prop­erty more gen­er­ally for all \( 0 < \alpha < 1 \).

This un­ex­pec­ted ap­pear­ance of size-biased sampling in the ana­lys­is of ex­cur­sion lengths of Browni­an mo­tion, and more gen­er­ally for any Markov pro­cess whose zero set is the range of a stable sub­or­din­at­or, led to a se­quence of pa­pers which ex­plored vari­ations and ex­ten­sions of this idea. The size factor pro­por­tion­al to \( t \) ap­pears al­most ac­ci­dent­ally in this ana­lys­is, as the ra­tio of the mass of the Lévy meas­ure to the right of \( t \) re­l­at­ive to the dens­ity of the Lévy meas­ure at \( t \): \( t^{-\alpha}/t^{-\alpha - 1} = t \). But size-biased sampling is an op­er­a­tion on in­ter­val par­ti­tions, or on as­so­ci­ated ran­dom dis­crete dis­tri­bu­tions, with nat­ur­al in­ter­pret­a­tions in oth­er con­texts, in par­tic­u­lar in Bayesian non­para­met­ric stat­ist­ics, in spe­cies sampling, in pop­u­la­tion ge­net­ics, and in mod­el­ing of time-evol­u­tions of ran­dom par­ti­tions. In such con­texts, it had been known for some time that it was of­ten easi­est to de­scribe a size-biased ran­dom per­muta­tion of lengths, defined by put­ting in­ter­val lengths in the or­der they are dis­covered in a pro­cess of sampling by a se­quence of in­de­pend­ent uni­form ran­dom points in the in­ter­val. This sug­ges­ted that it should be fruit­ful to de­scribe the size-biased ran­dom per­muta­tion of \( {\mathbf V}(T)/T \) de­rived as above from the range of a stable sub­or­din­at­or. This prob­lem was solved by Per­man–Pit­man–Yor [◊] for a gen­er­al sub­or­din­at­or, and led in the stable case to the two-para­met­er Pois­son–Di­rich­let dis­tri­bu­tion, de­noted by \( \operatorname{PD}(\alpha, \theta \)), stud­ied in de­tail in Pit­man–Yor [◊]. This is a prob­ab­il­ity dis­tri­bu­tion for a se­quence \( (V_n, n = 1,2, \ldots) \) with \( V_{1} > V_{2} > \cdots \) and \( \sum_{n}V_{n}=1 \). Let \( 0\leq \alpha < 1 \) and \( \theta > -\alpha \), and let \( W_n,\ n=1,2,\cdots, \) be in­de­pend­ent ran­dom vari­ables, with \( W_{n} \) hav­ing a beta(\( 1-\alpha,\theta+n\alpha \)) dis­tri­bu­tion. Put \[ \tilde{V}_{1}=W_{1}, \tilde{V}_{n}= (1-W_{1})\cdots (1-W_{n-1})W_{n},\quad n\geq 2, \] and let \( V_{1}\geq V_{2}\geq\cdots \) be the ranked val­ues of \( \tilde{V}_{1},\tilde{V}_{2},\cdots \). Then \( (\tilde{V}_{1},\tilde{V}_{2},\cdots) \) is a size-biased ran­dom per­muta­tion of \( (V_{n}) \), whose dis­tri­bu­tion is \( \operatorname{PD}(\alpha,\theta \)). In par­tic­u­lar, for \( 0 < \alpha < 1, \theta = 0 \), the \( \operatorname{PD}(\alpha,\theta \)) dis­tri­bu­tion is the dis­tri­bu­tion of ranked re­l­at­ive lengths \( {\mathbf V}(T)/T \) de­rived from the range of a stable\( (\alpha) \) sub­or­din­at­or \( (T_s) \), as above, either for \( T = t \) a fixed time, or for \( T= T_s \). And for \( 0 < \alpha < 1, \theta = \alpha \) this is the dis­tri­bu­tion of ranked re­l­at­ive lengths \( {\mathbf V}(G_T)/G_T \) in the same set­ting. When \( Z \) is the zero set of a Browni­an mo­tion or re­cur­rent \( \operatorname{BES}(\delta) \) pro­cess \( B \), with \( \delta = 2 - 2 \alpha \), the path of \( B \) on \( [0,G_T] \) can be res­caled to a stand­ard Browni­an or \( \operatorname{BES}(\delta) \) bridge of length 1, from 0 to 0. So \( \operatorname{PD}(\alpha,\alpha \)) de­scribes the dis­tri­bu­tion of ranked lengths of ex­cur­sions of a stand­ard Browni­an or Bessel bridge of di­men­sion \( 2 - 2 \alpha \). The key to this re­mark­ably simple re­la­tion between ranked lengths of ex­cur­sions of a stand­ard Browni­an or Bessel bridge and those of the cor­res­pond­ing un­con­di­tioned pro­cess is a ba­sic ab­so­lute condinu­ity re­la­tion, due to Bi­ane–Le Gall–Yor [◊] in the Browni­an case, whereby the the bridge is rep­res­en­ted by a change of meas­ure re­l­at­ive to the pro­cess ob­tained by Browni­an scal­ing of the un­con­di­tioned pro­cess up to an in­verse loc­al time. The dens­ity factor in­volved is just a con­stant times a power of the loc­al time at 0. Such changes of meas­ure provided a key to fur­ther ana­lys­is of the Pois­son–Di­rich­let fam­ily of ran­dom dis­crete dis­tri­bu­tions. For a re­view of de­vel­op­ments and ap­plic­a­tions of this fam­ily in vari­ous set­tings see Pit­man [◊]. The term Pit­man–Yor pro­cess was in­tro­duced by Ish­waran–James [◊] for a ran­dom meas­ure with \( \operatorname{PD}(\alpha,\theta) \) dis­trib­uted atoms. Such ran­dom meas­ures are now ap­plied in the set­ting of Bayesian non-para­met­ric stat­ist­ics and re­lated de­vel­op­ments in ma­chine learn­ing, to mod­el clus­ter­ing phe­nom­ena ex­hib­it­ing typ­ic­al power-law be­ha­vi­or en­countered in a num­ber of con­tem­por­ary ap­plic­a­tions. See [◊], [◊] for some re­cent de­vel­op­ments with ref­er­ences to earli­er work. The two-para­met­er Pois­son–Di­rich­let dis­tri­bu­tion has also been ap­plied in the con­text of ran­dom frag­ment­a­tion trees by [◊], and to frag­ment­a­tion-co­ales­cence pro­cesses by Ber­toin [◊].

Hand in hand with ex­cur­sion the­ory, a re­cur­ring theme of Marc’s work was the study of last exit times, es­pe­cially for path frag­ments of Browni­an mo­tion and one-di­men­sion­al dif­fu­sions. His early work on last exits re­lied on his balay­age for­mula for semi-mar­tin­gales, which is closely re­lated to the Azéma–Yor [◊] solu­tion of Skorok­hod’s prob­lem. This prob­lem is to find, for a suit­able prob­ab­il­ity dis­tri­bu­tion \( \mu \) on \( \mathbb{R} \), a stop­ping time \( T \) of Browni­an mo­tion \( (B_t, t \ge 0 ) \) such that \( B_T \) has dis­tri­bu­tion \( \mu \), and \( T \) is in some sense as small as pos­sible. The res­ult of Azéma–Yor is that if \( \mu \) has mean 0, and \( T_\mu:= \inf \{t: S_t \ge \psi_\mu(B_t) \} \), where \( \psi_\mu(x) \) is the mean of \( \mu \) con­di­tioned on \( [x,\infty) \), then \( B_{T_\mu} \) has dis­tri­bu­tion \( \mu \), and the mar­tin­gale \( (B_{T_\mu} \wedge t, t \ge 0) \) is uni­formly in­teg­rable. See Yen–Yor [◊], Sec­tions 3.4 and 3.6, for a re­view. Over time, Marc’s view of balay­age, last exits and the Azéma–Yor solu­tion of Skorok­hod’s prob­lem evolved to a point where these no­tions were fully in­teg­rated with Itô’s the­ory ap­plied to Browni­an ex­cur­sions, as in Obłój–Yor [◊]. See also Obłój [◊] for a sur­vey of the large num­ber of solu­tions to Skorok­hod’s prob­lem, amongst which the Azéma–Yor solu­tion stands out as both the most el­eg­ant and most ex­pli­cit.

In Pit­man–Yor [◊], The­or­em 6.1, we used mar­tin­gale cal­cu­lus to find a for­mula for the prob­ab­il­ity dens­ity of the last exit time of an up­wardly tran­si­ent dif­fu­sion pro­cess, and in­dic­ated a num­ber of ap­plic­a­tions to Bessel pro­cesses and prop­er­ties of Bessel func­tions. In prin­ciple, the idea is very simple. For a dis­crete time tran­si­ent Markov chain, with trans­ition prob­ab­il­it­ies \( p(t,x,y) \), every time the chain exits a state \( y \) could be the last time, if the pro­cess nev­er re­turns. So start­ing at \( x \) the prob­ab­il­ity that the last time in state \( y \) is at time \( t \) is \( p(t,x,y) \rho(y) \) where \( \rho(y) = P_y( T_y = \infty) \) for \( T_y \) the time of first re­turn to \( y \). For a con­tinu­ous time dif­fu­sion, the idea is that every time the pro­cess re­turns to \( y \) (now a con­tinuum of times) there is a risk of ul­ti­mately leav­ing \( y \) at rate \( \rho(y) \) per unit loc­al time at \( y \), where \( \rho(y) \) is the in­tens­ity of ex­cur­sions away from \( y \) with in­fin­ite life­time. This is in the ex­ten­sion of Itô’s the­ory to ex­cur­sions of a Markov pro­cess away from a tran­si­ent state, in­dic­ated by Mey­er [◊]. Con­sequently, un­der reg­u­lar­ity con­di­tions which en­sure that a dif­fu­sion pro­cess has a nice trans­ition dens­ity \( p(t,x,y) \) re­l­at­ive to its speed meas­ure \( m(dy) \), the prob­ab­il­ity dens­ity at time \( t \) of the last exit time of state \( y \) for the dif­fu­sion star­ted in state \( x \) is just \( p(t,x,y) \rho(y) \) for some \( \rho(y) \), which is just a nor­mal­iz­a­tion con­stant re­lated to the nor­mal­iz­a­tion of loc­al time at \( y \) and the rate of last exits from \( y \). Mar­tin­gale cal­cu­lus also al­lows \( \rho(y) \) to be ex­pressed in terms of the scale func­tion of the dif­fu­sion. In sub­sequent work, Marc re­turned of­ten to this theme, de­vel­op­ing some re­mark­able con­nec­tions between last exit times of Bessel pro­cesses and func­tion­als of geo­met­ric Browni­an mo­tion of in­terest in math­em­at­ic­al fin­ance. See for in­stance Yor [◊], [◊]. Start­ing from such ex­amples, and in­flu­enced by the gen­er­al the­ory of en­large­ment of filta­tions, which Marc de­veloped in col­lab­or­a­tion with Thi­erry Jeulin in the 1980’s, Marc and his stu­dents and col­lab­or­at­ors de­veloped a gen­er­al the­ory of last times defined as the ends of op­tion­al or pre­dic­able sets re­l­at­ive to a giv­en fil­tra­tion. They have also shown how this the­ory of last exit times has nat­ur­al in­ter­pret­a­tions and ap­plic­a­tions in math­em­at­ic­al fin­ance. See the text of Pro­feta–Roynette–Yor [◊], and the sur­vey of Nik­egh­bali–Platen [◊] for re­cent over­views of this work.

From “The Last Time” by Mick Jag­ger and Keith Richards (1965), a re­frain we en­joyed for many years, un­til fi­nally the last time came:

Well this could be the last time
This could be the last time
Maybe the last time
I don’t know. Oh no. Oh no