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 $\mathrm{BES}(3)$ bridge from 0 to 0 of length $t$. Here $\mathrm{BES}(\delta )$ for $\delta =1,2,\dots$ 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 $\mathrm{BES}(3)$ pro­cess. $\mathrm{BES}(3)$ 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 $\mathrm{\Lambda }$ on the space of ex­cur­sions, known as Itô’s ex­cur­sion law. For a re­flect­ing Browni­an mo­tion $X$ on $[0,\mathrm{\infty })$, there are the fol­low­ing two fun­da­ment­al de­scrip­tions of Itô’s law $\mathrm{\Lambda }$ 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 $(0,\mathrm{\infty })$; then giv­en $t$, run a $\mathrm{BES}(3)$ 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 $(0,\mathrm{\infty })$; then giv­en $m$, join back to back two in­de­pend­ent $\mathrm{BES}(3)$ 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 $\mathrm{BES}(3)$ is ex­ten­ded to one in­volving $\mathrm{BES}(\delta )$, 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 ${\mathrm{\Lambda }}_{00}^{\delta }$ 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 ${\mathrm{\Lambda }}_{00}^{\delta }$ is Itô’s ex­cur­sion law for ex­cur­sions of $\mathrm{BES}(4-\delta )$ 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 $${\mathrm{\Lambda }}_{00}^3=2n_{+}$$ and The­or­em 1 re­duces to the pre­ced­ing de­scrip­tions of Itô’s law $\mathrm{\Lambda }=2n_{+}$ for ex­cur­sions of a re­flect­ing Browni­an mo­tion.

For all $\delta \ge 2$ the meas­ure ${\mathrm{\Lambda }}_{00}^{\delta }$ 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 $(0,\mathrm{\infty })$ is a Lévy meas­ure only for $2<\delta <4$. So for $\delta \le 2$ or $\delta \ge 4$ the meas­ure ${\mathrm{\Lambda }}_{00}^{\delta }$ is not the ex­cur­sion law of any Markov pro­cess. Non­ethe­less, these meas­ures ${\mathrm{\Lambda }}_{00}^{\delta }$ 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 $2{\mathrm{\Lambda }}_{00}^4$ 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 $n_{+}=\frac{1}{2}{\mathrm{\Lambda }}_{00}^3$. 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 $\mathrm{BES}(\delta )$, and the meas­ure ${\mathrm{\Lambda }}_{00}^{\delta }$ 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 $\mathrm{BES}(\delta )$ 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 $(\mathrm{\Omega },\mathcal{F},P,({\mathcal{F}}_t{)}_{t\ge 0})$ sat­is­fy­ing some tech­nic­al con­di­tions, there is a $\sigma$-fi­nite meas­ure $Q$ on $(\mathrm{\Omega },\mathcal{F})$ 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 $\mathrm{BES}(\delta )$ bridge by its dens­ity on path space re­l­at­ive to the law ob­tained by tak­ing two in­de­pend­ent $\mathrm{BES}(\delta )$ 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 $\text{(1)}$, which arises with $F(R^{\mathrm{br}})$ and $F({\tilde{R}}^{\mathrm{br}})$ in $\text{(1)}$ 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 $[0,\mathrm{\infty })$ 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 $$\begin{array}{}\text{(4)}& \zeta (s):=\sum _{n=1}^{\mathrm{\infty }}{n}^{-s}(\mathrm{\Re }s>1)\end{array}$$ ad­mits a mero­morph­ic con­tinu­ation to the en­tire com­plex plane, with only a simple pole at 1, and that the func­tion $$\begin{array}{}\text{(5)}& \xi (s):=\frac{1}{2}s(s-1){\pi }^{-s/2}\mathrm{\Gamma }(\frac{1}{2}s)\zeta (s)(\mathrm{\Re }s>1)\end{array}$$ is the re­stric­tion to $(\mathrm{\Re }s>1)$ of a unique en­tire ana­lyt­ic func­tion $\xi$, which sat­is­fies the func­tion­al equa­tion $$\begin{array}{}\text{(6)}& \xi (s)=\xi (1-s)\end{array}$$ 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 $$\begin{array}{}\text{(7)}& E(Y^s)=2\xi (s)(s\in \mathbb{C})\end{array}$$ Bi­ane–Yor showed that Riemann’s func­tion­al equa­tion $\text{(6)}$ is a con­sequence of the agree­ment for­mula $\text{(1)}$ for the max­im­um for $\delta =3$, com­bined with Chung’s re­mark­able iden­tity in dis­tri­bu­tion that $$\begin{array}{}\text{(8)}& Y^2\stackrel{d}{=}\frac{\pi }{2}(T+\hat{T})\end{array}$$ for $T$ and $\hat{T}$ the re­spect­ive hit­ting times of 1 of two in­de­pend­ent in­de­pend­ent $\mathrm{BES}(3)$ 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 $$\mathbf{V}(t):=(V_1(t),V_2(t),\cdots ),V_1(t)\ge V_2(t)\ge \cdots ,$$ 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 $$\mathbf{V}(t)/t\stackrel{d}{=}\mathbf{V}(T_s)/T_s\text{for all}t>0\text{and}s>0.$$ In the case $\alpha =\frac{1}{2}$, 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 $$P(J=j|\mathbf{V}(T))=V_j(T)/T(j=1,2,\dots )$$ 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 $\mathrm{PD}(\alpha ,\theta$), stud­ied in de­tail in Pit­man–Yor [◊]. This is a prob­ab­il­ity dis­tri­bu­tion for a se­quence $(V_n,n=1,2,\dots )$ with $V_1>V_2>\cdots$ and $\sum _n{V}_n=1$. Let $0\le \alpha <1$ 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 $${\tilde{V}}_1=W_1,{\tilde{V}}_n=(1-W_1)\cdots (1-W_{n-1})W_n,n\ge 2,$$ and let $V_1\ge V_2\ge \cdots$ 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 $\mathrm{PD}(\alpha ,\theta$). In par­tic­u­lar, for $0<\alpha <1,\theta =0$, the $\mathrm{PD}(\alpha ,\theta$) 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 $\mathrm{BES}(\delta )$ pro­cess $B$, with $\delta =2-2\alpha$, the path of $B$ on $[0,G_T]$ can be res­caled to a stand­ard Browni­an or $\mathrm{BES}(\delta )$ bridge of length 1, from 0 to 0. So $\mathrm{PD}(\alpha ,\alpha$) 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 $\mathrm{PD}(\alpha ,\theta )$ 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 $[x,\mathrm{\infty })$, 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 $\rho (y)=P_y(T_y=\mathrm{\infty })$ 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