J. W. Pitman and M. Yor :
“Processus de Bessel, et mouvement brownien, avec ‘drift’ ”
[Bessel processes, and Brownian motion, with ‘drift’ ],
C. R. Acad. Sci., Paris, Sér. A
291 : 2
(1980 ),
pp. 151–153 .
MR
605004 Zbl
0438.60063 article
People
BibTeX
@article {key605004m,
AUTHOR = {Pitman, Jim W. and Yor, Marc},
TITLE = {Processus de {B}essel, et mouvement
brownien, avec ``drift'' [Bessel processes,
and {B}rownian motion, with ``drift'']},
JOURNAL = {C. R. Acad. Sci., Paris, S\'er. A},
FJOURNAL = {Comptes Rendus Hebdomadaires des S\'eances
de l'Acad\'emie des Sciences, S\'erie
A},
VOLUME = {291},
NUMBER = {2},
YEAR = {1980},
PAGES = {151--153},
NOTE = {MR:605004. Zbl:0438.60063.},
ISSN = {0151-0509},
}
J. Pitman and M. Yor :
“Bessel processes and infinitely divisible laws 285–370
in
Stochastic integrals
(Durham, UK, 7–17 July 1980 ).
Edited by D. Williams .
Lecture Notes in Mathematics 851 .
Springer (Berlin ),
1981 .
MR
620995 Zbl
0469.60076 incollection
People
BibTeX
@incollection {key620995m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Bessel processes and infinitely divisible
laws},
BOOKTITLE = {Stochastic integrals},
EDITOR = {Williams, David},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {851},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1981},
PAGES = {285--370},
DOI = {10.1007/BFb0088732},
NOTE = {(Durham, UK, 7--17 July 1980). MR:620995.
Zbl:0469.60076.},
ISSN = {0075-8434},
ISBN = {9783540106906},
}
J. Pitman and M. Yor :
“A decomposition of Bessel bridges Z. Wahrsch. Verw. Gebiete
59 : 4
(December 1982 ),
pp. 425–457 .
English translation of French original from Functional analysis in Markov processes (1982)
MR
656509 Zbl
0484.60062 article
People
BibTeX
@article {key656509m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {A decomposition of {B}essel bridges},
JOURNAL = {Z. Wahrsch. Verw. Gebiete},
FJOURNAL = {Zeitschrift f\"ur Wahrscheinlichkeitstheorie
und Verwandte Gebiete},
VOLUME = {59},
NUMBER = {4},
MONTH = {December},
YEAR = {1982},
PAGES = {425--457},
DOI = {10.1007/BF00532802},
NOTE = {English translation of French original
from \textit{Functional analysis in
Markov processes} (1982). MR:656509.
Zbl:0484.60062.},
ISSN = {0044-3719},
}
J. Pitman and M. Yor :
“Sur une décomposition des ponts de Bessel ”
[On a decomposition of Bessel bridges ],
pp. 276–285
in
Functional analysis in Markov processes
(Katata and Kyoto, Japan, 21–29 August 1981 ).
Edited by M. Fukushima .
Lecture Notes in Mathematics 923 .
Springer (Berlin ),
1982 .
An English translation was published in Z. Wahrsch. Verw. Gebiete 59 :4 (1982)
MR
661630 Zbl
0499.60082 incollection
People
BibTeX
@incollection {key661630m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Sur une d\'ecomposition des ponts de
{B}essel [On a decomposition of {B}essel
bridges]},
BOOKTITLE = {Functional analysis in {M}arkov processes},
EDITOR = {Fukushima, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {923},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1982},
PAGES = {276--285},
NOTE = {(Katata and Kyoto, Japan, 21--29 August
1981). An English translation was published
in \textit{Z. Wahrsch. Verw. Gebiete}
\textbf{59}:4 (1982). MR:661630. Zbl:0499.60082.},
ISSN = {0075-8434},
ISBN = {9783540391555},
}
J. W. Pitman and M. Yor :
“The asymptotic joint distribution of windings of planar Brownian motion Bull. Am. Math. Soc., New Ser.
10 : 1
(January 1984 ),
pp. 109–111 .
MR
722863 Zbl
0535.60073 article
People
BibTeX
@article {key722863m,
AUTHOR = {Pitman, J. W. and Yor, M.},
TITLE = {The asymptotic joint distribution of
windings of planar {B}rownian motion},
JOURNAL = {Bull. Am. Math. Soc., New Ser.},
FJOURNAL = {American Mathematical Society. Bulletin.
New Series},
VOLUME = {10},
NUMBER = {1},
MONTH = {January},
YEAR = {1984},
PAGES = {109--111},
DOI = {10.1090/S0273-0979-1984-15205-4},
NOTE = {MR:722863. Zbl:0535.60073.},
ISSN = {0273-0979},
}
J. Pitman and M. Yor :
“Asymptotic laws of planar Brownian motion Ann. Probab.
14 : 3
(1986 ),
pp. 733–779 .
A follow-up to this was published in Ann. Probab. 17 :3 (1989)
MR
841582 Zbl
0607.60070 article
Abstract
People
BibTeX
Recent results on the asymptotic distribution of winding and crossing numbers are presented as part of a larger framework of asymptotic laws for planar Brownian motion. The approach is via random time changes, martingale calculus, and excursion theory.
@article {key841582m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Asymptotic laws of planar {B}rownian
motion},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {14},
NUMBER = {3},
YEAR = {1986},
PAGES = {733--779},
DOI = {10.1214/aop/1176992436},
NOTE = {A follow-up to this was published in
\textit{Ann. Probab.} \textbf{17}:3
(1989). MR:841582. Zbl:0607.60070.},
ISSN = {0091-1798},
}
J. Pitman and M. Yor :
“Level crossings of a Cauchy process Ann. Probab.
14 : 3
(1986 ),
pp. 780–792 .
MR
841583 Zbl
0602.60059 article
Abstract
People
BibTeX
The asymptotic distribution as \( t\to\infty \) is obtained for the number of jumps of a symmetric Cauchy process across level \( x \) up to time \( t \) , jointly as \( x \) varies. This result is related to the asymptotic joint distribution of windings of a planar Brownian motion about several points.
@article {key841583m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Level crossings of a {C}auchy process},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {14},
NUMBER = {3},
YEAR = {1986},
PAGES = {780--792},
DOI = {10.1214/aop/1176992437},
URL = {https://www.jstor.org/stable/2244133},
NOTE = {MR:841583. Zbl:0602.60059.},
ISSN = {0091-1798},
}
J. W. Pitman and M. Yor :
“Some divergent integrals of Brownian motion 109–116
in
Analytic and geometric stochastics: Papers in honour of G. E. H. Reuter ,
published as Adv. Appl. Probab.
18 .
Issue edited by D. G. Kendall, J. F. C. Williams, and D. Williams .
Applied Probability Trust (Sheffield, UK ),
December 1986 .
Supplementary issue.
MR
868512 Zbl
0618.60074 incollection
Abstract
People
BibTeX
Let \( (X_t \) , \( t\geq 0) \) denote a two-dimensional Brownian motion starting from 0. If
\[ f:\mathbb{R}^2\to\mathbb{R}_+ \]
is a measurable function, which is integrable with respect to Lebesgue measure, then, for each \( \epsilon\in (0,1) \) , the integral
\[ \int_{\epsilon}^{1}ds\,f(X_{s}) \]
is almost surely finite. The asymptotic behaviour of the integral as \( \epsilon\to 0 \) is studied and, for some particular values of \( f \) , unusual limits in law are obtained.
@article {key868512m,
AUTHOR = {Pitman, J. W. and Yor, M.},
TITLE = {Some divergent integrals of {B}rownian
motion},
JOURNAL = {Adv. Appl. Probab.},
FJOURNAL = {Advances in Applied Probability},
VOLUME = {18},
MONTH = {December},
YEAR = {1986},
PAGES = {109--116},
URL = {https://www.jstor.org/stable/20528781},
NOTE = {\textit{Analytic and geometric stochastics:
{P}apers in honour of {G}.~{E}.~{H}.
{R}euter}. Issue edited by D. G. Kendall,
J. F. C. Williams,
and D. Williams. Supplementary
issue. MR:868512. Zbl:0618.60074.},
ISSN = {0001-8678},
}
K. Burdzy, J. W. Pitman, and M. Yor :
Some asymptotic laws for crossings and excursions .
Technical report 112 ,
Department of Statistics, UC-Berkeley ,
September 1987 .
Also published in Colloque Paul Lévy sur les processus stochastiques (1988)
techreport
People
BibTeX
@techreport {key46862063,
AUTHOR = {Burdzy, Krzysztof and Pitman, Jim W.
and Yor, Marc},
TITLE = {Some asymptotic laws for crossings and
excursions},
TYPE = {Technical Report},
NUMBER = {112},
INSTITUTION = {Department of Statistics, UC-Berkeley},
MONTH = {September},
YEAR = {1987},
NOTE = {Also published in \textit{Colloque Paul
L\'evy sur les processus stochastiques}
(1988).},
}
J. Pitman and M. Yor :
“Compléments à l’étude asymptotique des nombres de tours du mouvement brownien complexe autour d’un nombre fini de points ”
[Complements to the study of the winding numbers of complex Brownian motion around a finite set of points ],
C. R. Acad. Sci., Paris, Sér. I
305 : 17
(1987 ),
pp. 757–760 .
MR
921145 Zbl
0624.60088 article
People
BibTeX
@article {key921145m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Compl\'ements \`a l'\'etude asymptotique
des nombres de tours du mouvement brownien
complexe autour d'un nombre fini de
points [Complements to the study of
the winding numbers of complex {B}rownian
motion around a finite set of points]},
JOURNAL = {C. R. Acad. Sci., Paris, S\'er. I},
FJOURNAL = {Comptes Rendus des S\'eances de l'Acad\'emie
des Sciences. S\'erie I. Math\'ematique},
VOLUME = {305},
NUMBER = {17},
YEAR = {1987},
PAGES = {757--760},
NOTE = {MR:921145. Zbl:0624.60088.},
ISSN = {0249-6291},
}
K. Burdzy, J. W. Pitman, and M. Yor :
“Some asymptotic laws for crossings and excursions ,”
pp. 59–74
in
Colloque Paul Lévy sur les processus stochastiques
[Paul Lévy colloquium on stochastic processes ]
(Palaiseau, France, 22–26 June 1987 ).
Edited by M. Métivier .
Astérisque 157–158 .
Société Mathématique de France (Paris ),
1988 .
Also published as a 1988 UC-Berkeley technical report .
MR
976213 Zbl
0666.60070 incollection
People
BibTeX
@incollection {key976213m,
AUTHOR = {Burdzy, Krzysztof and Pitman, Jim W.
and Yor, Marc},
TITLE = {Some asymptotic laws for crossings and
excursions},
BOOKTITLE = {Colloque {P}aul {L}\'evy sur les processus
stochastiques [Paul {L}\'evy colloquium
on stochastic processes]},
EDITOR = {M\'etivier, Michel},
SERIES = {Ast\'erisque},
NUMBER = {157--158},
PUBLISHER = {Soci\'et\'e Math\'ematique de France},
ADDRESS = {Paris},
YEAR = {1988},
PAGES = {59--74},
NOTE = {(Palaiseau, France, 22--26 June 1987).
Also published as a 1988 UC-Berkeley
technical report. MR:976213. Zbl:0666.60070.},
ISSN = {0303-1179},
}
J. Pitman and M. Yor :
“Further asymptotic laws of planar Brownian motion Ann. Probab.
17 : 3
(1989 ),
pp. 965–1011 .
This was a follow-up to an article published in Ann. Probab. 14 :3 (1986)
MR
1009441 Zbl
0686.60085 article
People
BibTeX
@article {key1009441m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Further asymptotic laws of planar {B}rownian
motion},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {17},
NUMBER = {3},
YEAR = {1989},
PAGES = {965--1011},
DOI = {10.1214/aop/1176991253},
NOTE = {This was a follow-up to an article published
in \textit{Ann. Probab.} \textbf{14}:3
(1986). MR:1009441. Zbl:0686.60085.},
ISSN = {0091-1798},
}
M. Barlow, J. Pitman, and M. Yor :
“On Walsh’s Brownian motions 275–293
in
Séminaire de probabilités XXIII
[Twenty-third probability seminar ].
Edited by J. Azéma, P. A. Meyer, and M. Yor .
Lecture Notes in Mathematics 1372 .
Springer (Berlin ),
1989 .
MR
1022917 Zbl
0747.60072 incollection
People
BibTeX
@incollection {key1022917m,
AUTHOR = {Barlow, Martin and Pitman, Jim and Yor,
Marc},
TITLE = {On {W}alsh's {B}rownian motions},
BOOKTITLE = {S\'eminaire de probabilit\'es {XXIII}
[Twenty-third probability seminar]},
EDITOR = {Az\'ema, J. and Meyer, P. A. and Yor,
M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1372},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1989},
PAGES = {275--293},
DOI = {10.1007/BFb0083979},
URL = {http://www.numdam.org/item?id=SPS_1989__23__275_0},
NOTE = {MR:1022917. Zbl:0747.60072.},
ISSN = {0075-8434},
ISBN = {9783540511915},
}
M. Barlow, J. Pitman, and M. Yor :
“Une extension multidimensionnelle de la loi de l’arc sinus A multidimensional extension of the arcsine law ],
pp. 294–314
in
Séminaire de probabilités XXIII
[Twenty-third probability seminar ].
Edited by J. Azéma, P. A. Meyer, and M. Yor .
Lecture Notes in Mathematics 1372 .
Springer (Berlin ),
1989 .
MR
1022918 Zbl
0738.60072 incollection
People
BibTeX
@incollection {key1022918m,
AUTHOR = {Barlow, Martin and Pitman, Jim and Yor,
Marc},
TITLE = {Une extension multidimensionnelle de
la loi de l'arc sinus [A multidimensional
extension of the arcsine law]},
BOOKTITLE = {S\'eminaire de probabilit\'es {XXIII}
[Twenty-third probability seminar]},
EDITOR = {Az\'ema, J. and Meyer, P. A. and Yor,
M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1372},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1989},
PAGES = {294--314},
DOI = {10.1007/BFb0083980},
URL = {http://www.numdam.org/item?id=SPS_1989__23__294_0},
NOTE = {MR:1022918. Zbl:0738.60072.},
ISSN = {0075-8434},
ISBN = {9783540511915},
}
K. Burdzy, J. Pitman, and M. Yor :
“Brownian crossings between spheres J. Math. Anal. Appl.
148 : 1
(May 1990 ),
pp. 101–120 .
MR
1052048 Zbl
0713.60082 article
Abstract
People
BibTeX
@article {key1052048m,
AUTHOR = {Burdzy, Krzysztof and Pitman, Jim and
Yor, Marc},
TITLE = {Brownian crossings between spheres},
JOURNAL = {J. Math. Anal. Appl.},
FJOURNAL = {Journal of Mathematical Analysis and
Applications},
VOLUME = {148},
NUMBER = {1},
MONTH = {May},
YEAR = {1990},
PAGES = {101--120},
DOI = {10.1016/0022-247X(90)90031-A},
NOTE = {MR:1052048. Zbl:0713.60082.},
ISSN = {0022-247X},
}
M. Perman, J. Pitman, and M. Yor :
“Size-biased sampling of Poisson point processes and excursions Probab. Theory Relat. Fields
92 : 1
(March 1992 ),
pp. 21–39 .
MR
1156448 Zbl
0741.60037 article
Abstract
People
BibTeX
Some general formulae are obtained for size-biased sampling from a Poisson point process in an abstract space where the size of a point is defined by an arbitrary strictly positive function. These formulae explain why in certain cases (gamma and stable) the size-biased permutation of the normalized jumps of a subordinator can be represented by a stickbreaking (residual allocation) scheme defined by independent beta random variables. An application is made to length biased sampling of excursions of a Markov process away from a recurrent point of its statespace, with emphasis on the Brownian and Bessel cases when the associated inverse local time is a stable subordinator. Results in this case are linked to generalizations of the arcsine law for the fraction of time spent positive by Brownian motion.
@article {key1156448m,
AUTHOR = {Perman, Mihael and Pitman, Jim and Yor,
Marc},
TITLE = {Size-biased sampling of {P}oisson point
processes and excursions},
JOURNAL = {Probab. Theory Relat. Fields},
FJOURNAL = {Probability Theory and Related Fields},
VOLUME = {92},
NUMBER = {1},
MONTH = {March},
YEAR = {1992},
PAGES = {21--39},
DOI = {10.1007/BF01205234},
NOTE = {MR:1156448. Zbl:0741.60037.},
ISSN = {0178-8051},
}
J. Pitman and M. Yor :
“Arcsine laws and interval partitions derived from a stable subordinator Proc. London Math. Soc. (3)
65 : 2
(September 1992 ),
pp. 326–356 .
MR
1168191 Zbl
0769.60014 article
Abstract
People
BibTeX
Lévy discovered that the fraction of time a standard one-dimensional Brownian motion \( B \) spends positive before time \( t \) has arcsine distribution, both for \( t \) a fixed time when \( B_t \neq 0 \) almost surely, and for \( t \) an inverse local time, when \( B_t = 0 \) almost surely. This identity in distribution is extended from the fraction of time spent positive to a large collection of functionals derived from the lengths and signs of excursions of \( B \) away from 0. Similar identities in distribution are associated with any process whose zero set is the range of a stable subordinator, for instance a Bessel process of dimension \( d \) for \( 0 \lt d \lt 2 \) .
@article {key1168191m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Arcsine laws and interval partitions
derived from a stable subordinator},
JOURNAL = {Proc. London Math. Soc. (3)},
FJOURNAL = {Proceedings of the London Mathematical
Society. Third Series},
VOLUME = {65},
NUMBER = {2},
MONTH = {September},
YEAR = {1992},
PAGES = {326--356},
DOI = {10.1112/plms/s3-65.2.326},
NOTE = {MR:1168191. Zbl:0769.60014.},
ISSN = {0024-6115},
}
S. M. Kozlov, J. W. Pitman, and M. Yor :
“Brownian interpretations of an elliptic integral 83–95
in
Seminar on stochastic processes, 1991
(Los Angeles, 23–25 March 1991 ).
Edited by E. Çinlar, K. L. Chung, and M. J. Sharpe .
Progress in Probability 29 .
Birkhäuser (Boston, MA ),
1992 .
Proceedings dedicated to the memory of Steven Orey.
MR
1172145 Zbl
0765.60082 incollection
Abstract
People
BibTeX
This paper presents some interpretations in terms of Brownian motion of the Legendre first order elliptic integral
\[ \int_0^{z_0}\frac{dz}{\sqrt{(1-z^2)(1-k^2z^2)}}\,. \]
We express the probability that a complex valued Brownian motion hits one subinterval of the real line before another in terms of the Legendre elliptic integral. Then we find the asymptotic distribution of the Legendre integral along a Brownian path, and deduce asymptotic laws for looping numbers of the Brownian path on the associated Riemann surface.
@incollection {key1172145m,
AUTHOR = {Kozlov, S. M. and Pitman, J. W. and
Yor, M.},
TITLE = {Brownian interpretations of an elliptic
integral},
BOOKTITLE = {Seminar on stochastic processes, 1991},
EDITOR = {\c{C}inlar, Erhan and Chung, Kai Lai
and Sharpe, M. J.},
SERIES = {Progress in Probability},
NUMBER = {29},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1992},
PAGES = {83--95},
DOI = {10.1007/978-1-4612-0381-0_8},
NOTE = {(Los Angeles, 23--25 March 1991). Proceedings
dedicated to the memory of Steven Orey.
MR:1172145. Zbl:0765.60082.},
ISSN = {1050-6977},
ISBN = {9780817636289},
}
S. M. Kozlov, J. Pitman, and M. Yor :
“Wiener football Teor. Veroyatnost. i Primenen.
37 : 3
(1992 ),
pp. 562–564 .
An English translation was published in Theory Probab. Appl. 37 :3 (1992)Teor. Veroyatnost. i Primenen. 38 :1 (1993)
MR
1214362 article
People
BibTeX
@article {key1214362m,
AUTHOR = {Kozlov, S. M. and Pitman, Jim and Yor,
Marc},
TITLE = {Wiener football},
JOURNAL = {Teor. Veroyatnost. i Primenen.},
FJOURNAL = {Teoriya Veroyatnoste\u\i\ i e\"e Primeneniya.
Rossi\u\i skaya Akademiya Nauk.},
VOLUME = {37},
NUMBER = {3},
YEAR = {1992},
PAGES = {562--564},
URL = {http://mi.mathnet.ru/tvp4065},
NOTE = {An English translation was published
in \textit{Theory Probab. Appl.} \textbf{37}:3
(1992). A letter concerning this article
appeared in \textit{Teor. Veroyatnost.
i Primenen.} \textbf{38}:1 (1993). MR:1214362.},
ISSN = {0040-361X},
}
S. M. Kozlov, J. B. Pitman, and M. Yor :
“Wiener football Theory Probab. Appl.
37 : 3
(1992 ),
pp. 550–553 .
Russian original was published in Teor. Veroyatnost. i Primenen. 37 :3 (1992)
Zbl
0773.60079 article
Abstract
People
BibTeX
The paper suggests an abstract model of the game of soccer (football) on an infinite field, where the path of the ball is the planar Wiener trajectory. The asymptotical distribution of the soccer score is obtained for large time \( T \) . It shows that the variance of the score is proportional to \( (K^{\prime}/K)\log T \) , where \( K \) and \( K^{\prime} \) are the periods of an elliptic integral with singularities at the locations of the goal posts. In particular this relation shows how the score in this game depends on the width of the goals. It is interesting to note that today FIFA is considering a possible increase of the goal’s width in connection with the fall of attendance at soccer matches even during the world championships. Of course, the suggested model is one among many possible models taking more and more soccer rules into account. For instance, the slow logarithmic growth of the variance in the given model is a result of the unboundedness of the soccer field. A more realistic model of the same game would be on a rectangular field where the Wiener motion is reflected from the boundary along the normal; one may obtain the central limit theorem for the score in this model by analogy with [Kozlov 1985] since this is actually Wiener soccer on a torus due to the symmetry. In such a model the variance of the score increases as fast as \( T \) and the limiting distribution turns out to be Gaussian. However there is no such explicit formula for the limiting variance as we have in the suggested model of planar Wiener soccer.
@article {key0773.60079z,
AUTHOR = {Kozlov, S. M. and Pitman, J. B. and
Yor, M.},
TITLE = {Wiener football},
JOURNAL = {Theory Probab. Appl.},
FJOURNAL = {Theory of Probability and its Applications},
VOLUME = {37},
NUMBER = {3},
YEAR = {1992},
PAGES = {550--553},
DOI = {10.1137/1137106},
NOTE = {Russian original was published in \textit{Teor.
Veroyatnost. i Primenen.} \textbf{37}:3
(1992). Zbl:0773.60079.},
ISSN = {0040-585X},
}
J. W. Pitman and M. Yor :
“Dilatations d’espace-temps, réarrangements des trajectoires browniennes, et quelques extensions d’une identité de Knight ”
[Spacetime dilations, rearrangements of Brownian trajectories, and some extensions of an identity of Knight ],
C. R. Acad. Sci., Paris, Sér. I
316 : 7
(1993 ),
pp. 723–726 .
MR
1214423 Zbl
0789.60059 article
People
BibTeX
@article {key1214423m,
AUTHOR = {Pitman, James W. and Yor, Marc},
TITLE = {Dilatations d'espace-temps, r\'earrangements
des trajectoires browniennes, et quelques
extensions d'une identit\'e de {K}night
[Spacetime dilations, rearrangements
of {B}rownian trajectories, and some
extensions of an identity of {K}night]},
JOURNAL = {C. R. Acad. Sci., Paris, S\'er. I},
FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences.
S\'erie I. Math\'ematique},
VOLUME = {316},
NUMBER = {7},
YEAR = {1993},
PAGES = {723--726},
NOTE = {MR:1214423. Zbl:0789.60059.},
ISSN = {0764-4442},
}
P. Fitzsimmons, J. Pitman, and M. Yor :
“Markovian bridges: Construction, Palm interpretation, and splicing 101–134
in
Seminar on stochastic processes, 1992
(Seattle, WA, 26–28 March 1992 ).
Edited by E. Çinlar, K. L. Chung, and M. J. Sharpe .
Progress in Probability 33 .
Birkhäuser (Boston, MA ),
1993 .
MR
1278079 Zbl
0844.60054 incollection
People
BibTeX
@incollection {key1278079m,
AUTHOR = {Fitzsimmons, Pat and Pitman, Jim and
Yor, Marc},
TITLE = {Markovian bridges: {C}onstruction, {P}alm
interpretation, and splicing},
BOOKTITLE = {Seminar on stochastic processes, 1992},
EDITOR = {\c{C}inlar, E. and Chung, K. L. and
Sharpe, M. J.},
SERIES = {Progress in Probability},
NUMBER = {33},
PUBLISHER = {Birkh\"auser},
ADDRESS = {Boston, MA},
YEAR = {1993},
PAGES = {101--134},
DOI = {10.1007/978-1-4612-0339-1_5},
NOTE = {(Seattle, WA, 26--28 March 1992). MR:1278079.
Zbl:0844.60054.},
ISSN = {1050-6977},
ISBN = {9780817636494},
}
S. M. Kozlov :
“Letter to the editors: ‘Wiener soccer’ Teor. Veroyatnost. i Primenen.
38 : 1
(1993 ),
pp. 204 .
This concerns an article published in Teor. Veroyatnost. i Primenen. 37 :3 (1992)
MR
1317797 article
People
BibTeX
@article {key1317797m,
AUTHOR = {Kozlov, S. M.},
TITLE = {Letter to the editors: ``{W}iener soccer''},
JOURNAL = {Teor. Veroyatnost. i Primenen.},
FJOURNAL = {Rossi\u\i skaya Akademiya Nauk. Teoriya
Veroyatnoste\u\i\ i ee Primeneniya},
VOLUME = {38},
NUMBER = {1},
YEAR = {1993},
PAGES = {204},
URL = {http://mi.mathnet.ru/eng/tvp3889},
NOTE = {This concerns an article published in
\textit{Teor. Veroyatnost. i Primenen.}
\textbf{37}:3 (1992). MR:1317797.},
ISSN = {0040-361X},
}
J. Pitman and M. Yor :
“Random discrete distributions derived from self-similar random sets Electron. J. Probab.
1 : 4
(1996 ).
Article no. 4, 28 pp.
MR
1386296 Zbl
0891.60042 article
Abstract
People
BibTeX
A model is proposed for a decreasing sequence of random variables \( (V_1 \) , \( V_2 \) , \( \dots) \) with \( \sum_n V_n = 1 \) , which generalizes the Poisson–Dirichlet distribution and the distribution of ranked lengths of excursions of a Brownian motion or recurrent Bessel process. Let \( V_n \) be the length of the \( n \) th longest component interval of \( [0,1]\backslash Z \) , where \( Z \) is an a.s. non-empty random closed of \( (0,\infty) \) of Lebesgue measure 0, and \( Z \) is self-similar, i.e., \( cZ \) has the same distribution as \( Z \) for every \( c \geq 0 \) . Then for \( 0 \leq a \lt b \leq 1 \) the expected number of \( n \) ’s such that \( V_n \in (a,b) \) equals
\[ \int_a^b v^{-1} F(dv) \]
where the structural distribution \( F \) is identical to the distribution of
\[ 1-\sup(Z\cap [0,1]) .\]
Then \( F(dv) = f(v)\,dv \) where \( (1-v)f(v) \) is a decreasing function of \( v \) , and every such probability distribution \( F \) on \( [0,1] \) can arise from this construction.
@article {key1386296m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Random discrete distributions derived
from self-similar random sets},
JOURNAL = {Electron. J. Probab.},
FJOURNAL = {Electronic Journal of Probability},
VOLUME = {1},
NUMBER = {4},
YEAR = {1996},
DOI = {10.1214/EJP.v1-4},
NOTE = {Article no. 4, 28 pp. MR:1386296. Zbl:0891.60042.},
ISSN = {1083-6489},
}
J. W. Pitman and M. Yor :
“Quelques identités en loi pour les processus de Bessel ”
[Some identities in law for Bessel processes ],
pp. 249–276
in
Hommage à P. A. Meyer et J. Neveu
[In homage to P. A. Meyer and J. Neveu ].
Astérisque 236 .
Société Mathématique de France (Paris ),
1996 .
MR
1417987 Zbl
0863.60035 incollection
People
BibTeX
@incollection {key1417987m,
AUTHOR = {Pitman, J. W. and Yor, M.},
TITLE = {Quelques identit\'es en loi pour les
processus de {B}essel [Some identities
in law for {B}essel processes]},
BOOKTITLE = {Hommage \`a {P}.~{A}. {M}eyer et {J}.
{N}eveu [In homage to {P}.~{A}. {M}eyer
and {J}. {N}eveu]},
SERIES = {Ast\'erisque},
NUMBER = {236},
PUBLISHER = {Soci\'et\'e Math\'ematique de France},
ADDRESS = {Paris},
YEAR = {1996},
PAGES = {249--276},
NOTE = {MR:1417987. Zbl:0863.60035.},
ISSN = {0303-1179},
}
J. Pitman and M. Yor :
“Decomposition at the maximum for excursions and bridges of one-dimensional diffusions 293–310
in
Itô’s stochastic calculus and probability theory .
Edited by N. Ikeda .
Springer (Tokyo ),
1996 .
Book dedicated to dedicated to Kiyosi Itô on the occasion of his 80th birthday.
MR
1439532 Zbl
0877.60053 incollection
Abstract
People
BibTeX
In his fundamental paper [1971], 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 on the space of excursions, known as Itô’s excursion law. Accounts of Itô’s theory of excursions can now be found in several textbooks [Rogers and Williams 1987; Revuz and Yor 1994; Blumenthal 1992]. His theory has also been generalized to excursions of Markov processes away from a set of states [Maisonneuve 1975; Getoor and Sharpe 1982; Blumenthal 1992] and to excursions of stationary, not necessarily Markovian processes [Pitman 1986].
@incollection {key1439532m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Decomposition at the maximum for excursions
and bridges of one-dimensional diffusions},
BOOKTITLE = {It\^o's stochastic calculus and probability
theory},
EDITOR = {Ikeda, Nobuyuki},
PUBLISHER = {Springer},
ADDRESS = {Tokyo},
YEAR = {1996},
PAGES = {293--310},
DOI = {10.1007/978-4-431-68532-6_19},
NOTE = {Book dedicated to dedicated to Kiyosi
It\^o on the occasion of his 80th birthday.
MR:1439532. Zbl:0877.60053.},
ISBN = {9784431701866},
}
J. Pitman and M. Yor :
“The two-parameter Poisson–Dirichlet distribution derived from a stable subordinator Ann. Probab.
25 : 2
(1997 ),
pp. 855–900 .
MR
1434129 Zbl
0880.60076 article
Abstract
People
BibTeX
The two-parameter Poisson-Dirichlet distribution, denoted \( \textrm{PD}(\alpha,\theta) \) is a probability distribution on the set of decreasing positive sequences with sum 1. The usual Poisson–Dirichlet distribution with a single parameter \( \theta \) , introduced by Kingman, is \( \textrm{PD}(0,\theta) \) . Known properties of \( \textrm{PD}(0,\theta) \) , including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of \( \textrm{PD}(\alpha,\theta) \) is a simple residual allocation model proposed by Engen in the context of species diversity, and rediscovered by Perman and the authors in the study of excursions of Brownian motion and Bessel processes. For \( 0\lt \alpha\lt 1 \) , \( \textrm{PD}(\alpha,0) \) is the asymptotic distribution of ranked lengths of excursions of a Markov chain away from a state whose recurrence time distribution is in the domain of attraction of a stable law of index \( \alpha \) . Formulae in this case trace back to work of Darling, Lamperti and Wendel in the 1950s and 1960s. The distribution of ranked lengths of excursions of a one-dimensional Brownian motion is \( \textrm{PD}(1/2,0) \) , and the corresponding distribution for a Brownian bredge is \( \textrm{PD}(1/2,1/2) \) . The \( \textrm{PD}(\alpha,0) \) and \( \textrm{PD}(\alpha,\alpha) \) distributions admit a similar interpretation in terms of the ranked lengths of excursions of a semistable Markov process whose zero set is the range of a stable subordinator of index \( \alpha \) .
@article {key1434129m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {The two-parameter {P}oisson--{D}irichlet
distribution derived from a stable subordinator},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {25},
NUMBER = {2},
YEAR = {1997},
PAGES = {855--900},
DOI = {10.1214/aop/1024404422},
NOTE = {MR:1434129. Zbl:0880.60076.},
ISSN = {0091-1798},
}
M. Jeanblanc, J. Pitman, and M. Yor :
“The Feynman–Kac formula and decomposition of Brownian paths ,”
Comput. Appl. Math.
16 : 1
(1997 ),
pp. 27–52 .
MR
1458521 Zbl
0877.60027 article
Abstract
People
BibTeX
@article {key1458521m,
AUTHOR = {Jeanblanc, M. and Pitman, J. and Yor,
M.},
TITLE = {The {F}eynman--{K}ac formula and decomposition
of {B}rownian paths},
JOURNAL = {Comput. Appl. Math.},
FJOURNAL = {Computational and Applied Mathematics},
VOLUME = {16},
NUMBER = {1},
YEAR = {1997},
PAGES = {27--52},
NOTE = {MR:1458521. Zbl:0877.60027.},
ISSN = {0101-8205},
}
J. Pitman and M. Yor :
“On the lengths of excursions of some Markov processes 272–286
in
Séminaire de probabilités XXXI
[Thirty-first probability seminar ].
Edited by J. Azéma, M. Emery, and M. Yor .
Lecture Notes in Mathematics 1655 .
Springer (Berlin ),
1997 .
MR
1478737 Zbl
0884.60071 incollection
Abstract
People
BibTeX
Results are obtained regarding the distribution of the ranked lengths of component intervals in the complement of the random set of times when a recurrent Markov process returns to its starting point. Various martingales are described in terms of the Lévy measure of the Poisson point process of interval lengths on the local time scale. The martingales derived from the zero set of a one-dimensional diffusion are related to martingales studied by Azéma and Rainer. Formulae are obtained which show how the distribution of interval lengths is affected when the underlying process is subjected to a Girsanov transoformation. In particular, results for the zero set of an Ornstein–Uhlenbeck process or a Cox–Ingersoll–Ross process are derived from results for a Brownian motion or recurrent Bessel process, when the zero set is the range of a stable subordinator.
@incollection {key1478737m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {On the lengths of excursions of some
{M}arkov processes},
BOOKTITLE = {S\'eminaire de probabilit\'es {XXXI}
[Thirty-first probability seminar]},
EDITOR = {Az\'ema, J. and Emery, M. and Yor, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1655},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1997},
PAGES = {272--286},
DOI = {10.1007/BFb0119313},
URL = {http://www.numdam.org/item?id=SPS_1997__31__272_0},
NOTE = {MR:1478737. Zbl:0884.60071.},
ISSN = {0075-8434},
ISBN = {9783540626343},
}
J. Pitman and M. Yor :
“On the relative lengths of excursions derived from a stable subordinator 287–305
in
Séminaire de probabilités XXXI
[Thirty-first probability seminar ].
Edited by J. Azéma, M. Emery, and M. Yor .
Lecture Notes in Mathematics 1655 .
Springer (Berlin ),
1997 .
MR
1478738 Zbl
0884.60072 incollection
People
BibTeX
@incollection {key1478738m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {On the relative lengths of excursions
derived from a stable subordinator},
BOOKTITLE = {S\'eminaire de probabilit\'es {XXXI}
[Thirty-first probability seminar]},
EDITOR = {Az\'ema, J. and Emery, M. and Yor, M.},
SERIES = {Lecture Notes in Mathematics},
NUMBER = {1655},
PUBLISHER = {Springer},
ADDRESS = {Berlin},
YEAR = {1997},
PAGES = {287--305},
DOI = {Results are obtained concerning the
distribution of ranked relative lengths
of excursions of a recurrent Markov
process from a point in its state space
whose inverse local time process is
a stable subordinator. It is shown that
for a large class of random times \$T\$
the distribution of relative excursion
lengths prior to \$T\$ is the same as
if \$T\$ were a fixed time. It follows
that the generalized arc-sine laws of
Lamperti extend to such random times
\$T\$. For some other random times \$T\$,
absolute continuity relations are obtained
which relate the law of the relative
lengths at time \$T\$ to the law at a
fixed time.},
URL = {http://www.numdam.org/item?id=SPS_1997__31__287_0},
NOTE = {MR:1478738. Zbl:0884.60072.},
ISSN = {0075-8434},
ISBN = {9783540626343},
}
J. Pitman and M. Yor :
“Ranked functionals of Brownian excursions C. R. Acad. Sci., Paris, Sér. I
326 : 1
(1998 ),
pp. 93–97 .
With French summary.
MR
1649517 Zbl
0924.60073 article
Abstract
People
BibTeX
It was shown, in our previous work, that the law of the sequence of normalized ranked lengths of Brownian excursions considered up to a random time \( T \) is the same for a large class of random times \( T \) .
We present now some results about (unnormalized) ranked heights of Brownian excursions, which although quite different from those obtained for the lengths, have led us to extend the scope of both studies.
@article {key1649517m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Ranked functionals of {B}rownian excursions},
JOURNAL = {C. R. Acad. Sci., Paris, S\'er. I},
FJOURNAL = {Comptes Rendus de l'Acad\'emie des Sciences.
S\'erie I. Math\'ematique},
VOLUME = {326},
NUMBER = {1},
YEAR = {1998},
PAGES = {93--97},
DOI = {10.1016/S0764-4442(97)82719-X},
NOTE = {With French summary. MR:1649517. Zbl:0924.60073.},
ISSN = {0764-4442},
}
J. Pitman and M. Yor :
“Random Brownian scaling identities and splicing of Bessel processes Ann. Probab.
26 : 4
(1998 ),
pp. 1683–1702 .
MR
1675059 Zbl
0937.60079 article
Abstract
People
BibTeX
An identity in distribution due to Knight for Brownian motion is extended in two different ways: first by replacing the supremum of a reflecting Brownian motion by the range of an unreflected Brownian motion and second by replacing the reflecting Brownian motion by a recurrent Bessel process. Both extensions are explained in terms of random Brownian scaling transformations and Brownian excursions. The first extension is related to two different constructions of Itô’s law of Brownian excursions, due to Williams and Bismut, each involving back-to-back splicing of fragments of two independent three-dimensional Bessel processes. Generalizations of both splicing constructions are described, which involve Bessel processes and Bessel bridges of arbitrary positive real dimension.
@article {key1675059m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Random {B}rownian scaling identities
and splicing of {B}essel processes},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {26},
NUMBER = {4},
YEAR = {1998},
PAGES = {1683--1702},
DOI = {10.1214/aop/1022855878},
NOTE = {MR:1675059. Zbl:0937.60079.},
ISSN = {0091-1798},
}
J. Pitman and M. Yor :
Laws of homogeneous functionals of Brownian motion ,
1998 .
unpublished
People
BibTeX
@unpublished {key55776024,
AUTHOR = {Pitman, Jim and Yor, M.},
TITLE = {Laws of homogeneous functionals of {B}rownian
motion},
YEAR = {1998},
}
J. Pitman and M. Yor :
“Laplace transforms related to excursions of a one-dimensional diffusion Bernoulli
5 : 2
(1999 ),
pp. 249–255 .
MR
1681697 Zbl
0921.60015 article
Abstract
People
BibTeX
@article {key1681697m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Laplace transforms related to excursions
of a one-dimensional diffusion},
JOURNAL = {Bernoulli},
FJOURNAL = {Bernoulli. Official Journal of the Bernoulli
Society for Mathematical Statistics
and Probability},
VOLUME = {5},
NUMBER = {2},
YEAR = {1999},
PAGES = {249--255},
DOI = {10.2307/3318434},
NOTE = {MR:1681697. Zbl:0921.60015.},
ISSN = {1350-7265},
}
J. Pitman and M. Yor :
“The law of the maximum of a Bessel bridge Electron. J. Probab.
4
(1999 ).
Article no. 15, 35 pp.
MR
1701890 Zbl
0943.60084 article
Abstract
People
BibTeX
Let \( M_d \) be the maximum of a standard Bessel bridge of dimension \( d \) . A series formula for \( P(M_d\leq a) \) due to Gikhman and Kiefer for \( d= 1 \) , \( 2,\dots \) is shown to be valid for all real \( d\gt 0 \) . Various other characterizations of the distribution of \( M_d \) are given, including formulae for its Mellin transform, which is an entire function. The asymptotic distribution of \( M_d \) is described both as \( d \) tends to infinity and as \( d \) tends to zero.
@article {key1701890m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {The law of the maximum of a {B}essel
bridge},
JOURNAL = {Electron. J. Probab.},
FJOURNAL = {Electronic Journal of Probability},
VOLUME = {4},
YEAR = {1999},
URL = {https://projecteuclid.org/euclid.ejp/1457125524},
NOTE = {Article no. 15, 35 pp. MR:1701890. Zbl:0943.60084.},
ISSN = {1083-6489},
}
J. Pitman and M. Yor :
“Path decompositions of a Brownian bridge related to the ratio of its maximum and amplitude ,”
Studia Sci. Math. Hung.
35 : 3–4
(1999 ),
pp. 457–474 .
MR
1761927 Zbl
0973.60082 article
Abstract
People
BibTeX
We give two new proofs of Csáki’s formula for the law of the ratio \( 1-Q \) of the maximum relative to the amplitude (i.e., the maximum minus minimum) for a standard Brownian bridge. The second of these proofs is based on an absolute continuity relation between the law of the Brownian bridge restricted to the event (\( Q\leq v \) ) and the law of a process obtained by a Brownian scaling operation after back-to back joining of two independent three-dimensional Bessel processes, each started at \( v \) and run until it first hits 1. Variants of this construction and some properties of the joint law of \( Q \) and the amplitude are described.
@article {key1761927m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Path decompositions of a {B}rownian
bridge related to the ratio of its maximum
and amplitude},
JOURNAL = {Studia Sci. Math. Hung.},
FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica.
A Quarterly of the Hungarian Academy
of Sciences},
VOLUME = {35},
NUMBER = {3--4},
YEAR = {1999},
PAGES = {457--474},
NOTE = {MR:1761927. Zbl:0973.60082.},
ISSN = {0081-6906},
}
P. Carmona, F. Petit, J. Pitman, and M. Yor :
“On the laws of homogeneous functionals of the Brownian bridge ,”
Studia Sci. Math. Hung.
35 : 3–4
(1999 ),
pp. 445–455 .
MR
1762255 Zbl
0980.60099 article
Abstract
People
BibTeX
@article {key1762255m,
AUTHOR = {Carmona, P. and Petit, F. and Pitman,
J. and Yor, M.},
TITLE = {On the laws of homogeneous functionals
of the {B}rownian bridge},
JOURNAL = {Studia Sci. Math. Hung.},
FJOURNAL = {Studia Scientiarum Mathematicarum Hungarica.
A Quarterly of the Hungarian Academy
of Sciences},
VOLUME = {35},
NUMBER = {3--4},
YEAR = {1999},
PAGES = {445--455},
NOTE = {MR:1762255. Zbl:0980.60099.},
ISSN = {0081-6906},
}
J. Pitman and M. Yor :
“On the distribution of ranked heights of excursions of a Brownian bridge Ann. Probab.
29 : 1
(2001 ),
pp. 361–384 .
MR
1825154 Zbl
1033.60050 article
Abstract
People
BibTeX
The distribution of the sequence of ranked maximum and minimum values attained during excursions of a standard Brownian bridge \( (B^{\mathrm{br}}_t \) , \( 0\leq t\leq 1) \) is described. The height \( M^{\mathrm{br}+}_j \) of the \( j \) -th highest maximum over a positive excursion of the bridge has the same distribution as \( M^{\mathrm{br}+1}_1/j \) , where the distribution of
\[ M^{\mathrm{br}+1}_1=\sup_{0\lt t\lt 1}B^{\mathrm{br}}_t \]
is given by Lévy’s formula
\[ P(M^{\mathrm{br}+}_1 \gt x) = e^{-2x^2} .\]
The probability density of the height \( M^{\mathrm{br}}_j \) of the \( j \) th highest maximum of excursions of the reflecting Brownian bridge \( (|B^{\mathrm{br}}_t| \) , \( 0\leq t\leq 1) \) is given by a modification of the known \( \theta \) -function series for the density of
\[ M^{\mathrm{br}}_1 = \sup_{0\leq t\leq 1}|B^{\mathrm{br}}_t| .\]
These results are obtained from a more general description of the distribution of ranked values of a homogeneous functional of excursions of the standardized bridge of a self-similar recurrent Markov process.
@article {key1825154m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {On the distribution of ranked heights
of excursions of a {B}rownian bridge},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {29},
NUMBER = {1},
YEAR = {2001},
PAGES = {361--384},
DOI = {10.1214/aop/1008956334},
NOTE = {MR:1825154. Zbl:1033.60050.},
ISSN = {0091-1798},
}
R. Pemantle, Y. Peres, J. Pitman, and M. Yor :
“Where did the Brownian particle go? Electron. J. Probab.
6
(2001 ).
Article no. 10, 22 pp.
MR
1831805 Zbl
0977.60071 ArXiv
math/0404097 article
Abstract
People
BibTeX
Consider the radial projection onto the unit sphere of the path a \( d \) -dimensional Brownian motion \( W \) , started at the center of the sphere and run for unit time. Given the occupation measure \( \mu \) of this projected path, what can be said about the terminal point \( W(1) \) , or about the range of the original path? In any dimension, for each Borel set \( A \) in \( S^{d-1} \) , the conditional probability that the projection of \( W(1) \) is in \( A \) given \( \mu(A) \) is just \( \mu(A) \) . Nevertheless, in dimension \( d\geq 3 \) , both the range and the terminal point of \( W \) can be recovered with probability 1 from \( \mu \) . In particular, for \( d\geq 3 \) the conditional law of the projection of \( W(1) \) given \( \mu \) is not \( \mu \) . In dimension 2 we conjecture that the projection of \( W(1) \) cannot be recovered almost surely from \( \mu \) , and show that the conditional law of the projection of \( W(1) \) given \( \mu \) is not \( \mu \) .
@article {key1831805m,
AUTHOR = {Pemantle, Robin and Peres, Yuval and
Pitman, Jim and Yor, Marc},
TITLE = {Where did the {B}rownian particle go?},
JOURNAL = {Electron. J. Probab.},
FJOURNAL = {Electronic Journal of Probability},
VOLUME = {6},
YEAR = {2001},
DOI = {10.1214/EJP.v6-83},
NOTE = {Article no. 10, 22 pp. ArXiv:math/0404097.
MR:1831805. Zbl:0977.60071.},
ISSN = {1083-6489},
}
P. Biane, J. Pitman, and M. Yor :
“Probability laws related to the Jacobi theta and Riemann zeta functions, and Brownian excursions Bull. Am. Math. Soc. (N.S.)
38 : 4
(2001 ),
pp. 435–465 .
MR
1848256 Zbl
1040.11061 ArXiv
math.PR/9912170 article
Abstract
People
BibTeX
This paper reviews known results which connect Riemann’s integral representations of his zeta function, involving Jacobi’s theta function and its derivatives, to some particular probability laws governing sums of independent exponential variables. These laws are related to one-dimensional Brownian motion and to higher dimensional Bessel processes. We present some characterizations of these probability laws, and some approximations of Riemann’s zeta function which are related to these laws.
@article {key1848256m,
AUTHOR = {Biane, Philippe and Pitman, Jim and
Yor, Marc},
TITLE = {Probability laws related to the {J}acobi
theta and {R}iemann zeta functions,
and {B}rownian excursions},
JOURNAL = {Bull. Am. Math. Soc. (N.S.)},
FJOURNAL = {American Mathematical Society. Bulletin.
New Series},
VOLUME = {38},
NUMBER = {4},
YEAR = {2001},
PAGES = {435--465},
DOI = {10.1090/S0273-0979-01-00912-0},
NOTE = {ArXiv:math.PR/9912170. MR:1848256.
Zbl:1040.11061.},
ISSN = {0273-0979},
}
M. Jeanblanc, J. Pitman, and M. Yor :
“Self-similar processes with independent increments associated with Lévy and Bessel processes Stochastic Process. Appl.
100 : 1–2
(2002 ),
pp. 223–231 .
MR
1919614 Zbl
1059.60052 article
Abstract
People
BibTeX
Wolfe and Sato gave two different representations of a random variable \( X \) with a self-decomposable distribution in terms of processes with independent increments. This paper shows how either of these representations follows easily from the other, and makes these representations more explicit when \( X \) is either a first or last passage time for a Bessel process.
@article {key1919614m,
AUTHOR = {Jeanblanc, M. and Pitman, J. and Yor,
M.},
TITLE = {Self-similar processes with independent
increments associated with {L}\'evy
and {B}essel processes},
JOURNAL = {Stochastic Process. Appl.},
FJOURNAL = {Stochastic Processes and their Applications},
VOLUME = {100},
NUMBER = {1--2},
YEAR = {2002},
PAGES = {223--231},
DOI = {10.1016/S0304-4149(02)00098-4},
NOTE = {MR:1919614. Zbl:1059.60052.},
ISSN = {0304-4149},
}
J. Pitman and M. Yor :
“Hitting, occupation and inverse local times of one-dimensional diffusions: Martingale and excursion approaches Bernoulli
9 : 1
(2003 ),
pp. 1–24 .
MR
1963670 Zbl
1024.60032 article
Abstract
People
BibTeX
Basic relations between the distributions of hitting, occupation and inverse local times of a one-dimensional diffusion process \( X \) , first discussed by Itô and McKean, are reviewed from the perspectives of martingale calculus and excursion theory. These relations, and the technique of conditioning on \( L^y_T \) , the local time of \( X \) at level \( y \) before a suitable random time \( T \) , yield formulae for the joint Laplace transform of \( L^y_T \) and the times spent by \( X \) above and below level \( y \) up to time \( T \) .
@article {key1963670m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Hitting, occupation and inverse local
times of one-dimensional diffusions:
{M}artingale and excursion approaches},
JOURNAL = {Bernoulli},
FJOURNAL = {Bernoulli. Official Journal of the Bernoulli
Society for Mathematical Statistics
and Probability},
VOLUME = {9},
NUMBER = {1},
YEAR = {2003},
PAGES = {1--24},
DOI = {10.3150/bj/1068129008},
NOTE = {MR:1963670. Zbl:1024.60032.},
ISSN = {1350-7265},
}
J. Pitman and M. Yor :
“Infinitely divisible laws associated with hyperbolic functions Canad. J. Math.
55 : 2
(2003 ),
pp. 292–330 .
MR
1969794 Zbl
1039.11054 article
Abstract
People
BibTeX
The infinitely divisible distributions on \( \mathbb{R}^+ \) of random variables \( C_t \) , \( S_t \) and \( T_t \) , with Laplace transforms
\[ \Bigl(\frac{1}{\cosh\sqrt{2\lambda}}\Bigr)^t, \quad \Bigl(\frac{\sqrt{2\lambda}}{\sinh\sqrt{2\lambda}}\Bigr)^t, \quad \textrm{and} \quad \Bigl(\frac{\tanh\sqrt{2\lambda}}{\sqrt{2\lambda}}\Bigr)^t \]
respectively, are characterized for various \( t \gt 0 \) in a number of different ways: by simple relations between their moments and cumulants, by corresponding relations between the distributions and their Lévy measures, by recursions for their Mellin transforms, and by differential equations satisfied by their Laplace transforms. Some of these results are interpreted probabilistically via known appearances of these distributions for \( t=1 \) or 2 in the description of the laws of various functionals of Brownian motion and Bessel processes, such as the heights and lengths of excursions of a one-dimensional Brownian motion. The distributions of \( C_1 \) and \( S_2 \) are also known to appear in the Mellin representations of two important functions in analytic number theory, the Riemann zeta function and the Dirichlet \( L \) -function associated with the quadratic character modulo 4. Related families of infinitely divisible laws, including the gamma, logistic and generalized hyperbolic secant distributions, are derived from \( S_t \) and \( C_t \) by operations such as Brownian subordination, exponential tilting, and weak limits, and characterized in various ways.
@article {key1969794m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Infinitely divisible laws associated
with hyperbolic functions},
JOURNAL = {Canad. J. Math.},
FJOURNAL = {Canadian Journal of Mathematics},
VOLUME = {55},
NUMBER = {2},
YEAR = {2003},
PAGES = {292--330},
DOI = {10.4153/CJM-2003-014-x},
NOTE = {MR:1969794. Zbl:1039.11054.},
ISSN = {0008-414X},
}
J. Pitman and M. Yor :
Polynomials associated with Lévy processes ,
2003 .
unpublished
People
BibTeX
@unpublished {key56366981,
AUTHOR = {Pitman, J. and Yor, M.},
TITLE = {Polynomials associated with {L}\'evy
processes},
YEAR = {2003},
}
J. Pitman and M. Yor :
“Some properties of the arc-sine law related to its invariance under a family of rational maps 126–137
in
A festschrift for Herman Rubin .
Edited by A. Dasgupta .
IMS Lecture Notes Monograph Series 45 .
Institute of Mathematical Statistics (Beachwood, OH ),
2004 .
MR
2126891 Zbl
1268.37071 incollection
Abstract
People
BibTeX
This paper shows how the invariance of the arc-sine distribution on \( (0,1) \) under a family of rational maps is related on the one hand to various integral identities with probabilistic interpretations involving random variables derived from Brownian motion with arc-sine, Gaussian, Cauchy and other distributions, and on the other hand to results in the analytic theory of iterated rational maps.
@incollection {key2126891m,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {Some properties of the arc-sine law
related to its invariance under a family
of rational maps},
BOOKTITLE = {A festschrift for {H}erman {R}ubin},
EDITOR = {Dasgupta, Anirban},
SERIES = {IMS Lecture Notes Monograph Series},
NUMBER = {45},
PUBLISHER = {Institute of Mathematical Statistics},
ADDRESS = {Beachwood, OH},
YEAR = {2004},
PAGES = {126--137},
DOI = {10.1214/lnms/1196285384},
NOTE = {MR:2126891. Zbl:1268.37071.},
ISSN = {0749-2170},
ISBN = {9780940600614},
}
A. Gnedin, J. Pitman, and M. Yor :
“Asymptotic laws for compositions derived from transformed subordinators Ann. Probab.
34 : 2
(2006 ),
pp. 468–492 .
MR
2223948 Zbl
1142.60327 ArXiv
math/0403438 article
Abstract
People
BibTeX
A random composition of \( n \) appears when the points of a random closed set \( \tilde{\mathscr{R}}\subset [0,1] \) are used to separate into blocks \( n \) points sampled from the uniform distribution. We study the number of parts \( K_n \) of this composition and other related functionals under the assumption that
\[ \tilde{\mathscr{R}} = \phi(S_{\bullet}) ,\]
where \( (S_t \) , \( t\geq 0) \) is a subordinator and \( \phi:[0,\infty]\to [0,1] \) is a diffeomorphism. We derive the asymptotics of \( K_n \) when the Lévy measure of the subordinator is regularly varying at 0 with positive index. Specializing to the case of exponential function \( \phi(x) = 1-e^{-x} \) , we establish a connection between the asymptotics of \( K_n \) and the exponential functional of the subordinator.
@article {key2223948m,
AUTHOR = {Gnedin, Alexander and Pitman, Jim and
Yor, Marc},
TITLE = {Asymptotic laws for compositions derived
from transformed subordinators},
JOURNAL = {Ann. Probab.},
FJOURNAL = {The Annals of Probability},
VOLUME = {34},
NUMBER = {2},
YEAR = {2006},
PAGES = {468--492},
DOI = {10.1214/009117905000000639},
NOTE = {ArXiv:math/0403438. MR:2223948. Zbl:1142.60327.},
ISSN = {0091-1798},
}
A. Gnedin, J. Pitman, and M. Yor :
“Asymptotic laws for regenerative compositions: Gamma subordinators and the like Probab. Theory Relat. Fields
135 : 4
(August 2006 ),
pp. 576–602 .
MR
2240701 Zbl
1099.60023 ArXiv
math.PR/0405440 article
Abstract
People
BibTeX
For \( \tilde{\mathcal{R}} = 1 - \exp(-\mathcal{R}) \) a random closed set obtained by exponential transformation of the closed range \( \mathcal{R} \) of a subordinator, a regenerative composition of generic positive integer \( n \) is defined by recording the sizes of clusters of \( n \) uniform random points as they are separated by the points of \( \tilde{\mathcal{R}} \) . We focus on the number of parts \( K_n \) of the composition when \( \tilde{\mathcal{R}} \) is derived from a gamma subordinator. We prove logarithmic asymptotics of the moments and central limit theorems for \( K_n \) and other functionals of the composition such as the number of singletons, doubletons, etc. This study complements our previous work on asymptotics of these functionals when the tail of the Lévy measure is regularly varying at \( 0+ \) .
@article {key2240701m,
AUTHOR = {Gnedin, Alexander and Pitman, Jim and
Yor, Marc},
TITLE = {Asymptotic laws for regenerative compositions:
{G}amma subordinators and the like},
JOURNAL = {Probab. Theory Relat. Fields},
FJOURNAL = {Probability Theory and Related Fields},
VOLUME = {135},
NUMBER = {4},
MONTH = {August},
YEAR = {2006},
PAGES = {576--602},
DOI = {10.1007/s00440-005-0473-0},
NOTE = {ArXiv:math.PR/0405440. MR:2240701.
Zbl:1099.60023.},
ISSN = {0178-8051},
}
J. Pitman and M. Yor :
“Itô’s excursion theory and its applications Japan. J. Math. (3)
2 : 1
(March 2007 ),
pp. 83–96 .
MR
2295611 Zbl
1156.60066 article
Abstract
People
BibTeX
@article {key2295611m,
AUTHOR = {Pitman, J. and Yor, M.},
TITLE = {It\^o's excursion theory and its applications},
JOURNAL = {Japan. J. Math. (3)},
FJOURNAL = {Japanese Journal of Mathematics. 3rd
Series},
VOLUME = {2},
NUMBER = {1},
MONTH = {March},
YEAR = {2007},
PAGES = {83--96},
DOI = {10.1007/s11537-007-0661-z},
NOTE = {MR:2295611. Zbl:1156.60066.},
ISSN = {0289-2316},
}
J.-F. Le Gall and J. Pitman :
“Obituary: Marc Yor 1949–2014 Notices Am. Math. Soc.
61 : 5
(May 2014 ),
pp. 508–509 .
article
People
BibTeX
@article {key61562309,
AUTHOR = {Le Gall, Jean-Fran\c{c}ois and Pitman,
Jim},
TITLE = {Obituary: {M}arc {Y}or 1949--2014},
JOURNAL = {Notices Am. Math. Soc.},
FJOURNAL = {Notices of the American Mathematical
Society},
VOLUME = {61},
NUMBER = {5},
MONTH = {May},
YEAR = {2014},
PAGES = {508--509},
DOI = {10.1090/noti1128},
ISSN = {0002-9920},
}
J.-F. Le Gall and J. Pitman :
“Marc Yor (1949–2014) Notices Am. Math. Soc.
61 : 5
(May 2014 ),
pp. 508–509 .
MR
3203242 Zbl
1338.01050 article
People
BibTeX
@article {key3203242m,
AUTHOR = {Le Gall, Jean-Fran\c{c}ois and Pitman,
Jim},
TITLE = {Marc {Y}or (1949--2014)},
JOURNAL = {Notices Am. Math. Soc.},
FJOURNAL = {Notices of the American Mathematical
Society},
VOLUME = {61},
NUMBER = {5},
MONTH = {May},
YEAR = {2014},
PAGES = {508--509},
DOI = {10.1090/noti1128},
NOTE = {MR:3203242. Zbl:1338.01050.},
ISSN = {0002-9920},
}
J. Pitman :
“Marc Yor and Brownian excursions ,”
pp. 55–66
in
Marc Yor: La passion du mouvement brownien
[Marc Yor: The passion of Brownian motion ].
Edited by J. Bertoin, M. Jeanblanc, J.-F. Le Gall, and Z. Shi .
Société Mathématique de France (Paris ),
2015 .
Gazette des Mathématiciens and Matapli special issue.
incollection
People
BibTeX
@incollection {key43568391,
AUTHOR = {Pitman, Jim},
TITLE = {Marc {Y}or and {B}rownian excursions},
BOOKTITLE = {Marc {Y}or: {L}a passion du mouvement
brownien [Marc {Y}or: {T}he passion
of {B}rownian motion]},
EDITOR = {Bertoin, Jean and Jeanblanc, Monique
and Le Gall, Jean-Fran\c{c}ois and Shi,
Zhan},
PUBLISHER = {Soci\'et\'e Math\'ematique de France},
ADDRESS = {Paris},
YEAR = {2015},
PAGES = {55--66},
NOTE = {Gazette des Math\'ematiciens and Matapli
special issue.},
ISBN = {9782856298015},
}
J. Pitman and M. Yor :
A guide to Brownian motion stochastic processes .
Preprint ,
February 2018 .
ArXiv
1802.09679 techreport
Abstract
People
BibTeX
This is a guide to the mathematical theory of Brownian motion and related stochastic processes, with indications of how this theory is related to other branches of mathematics, most notably the classical theory of partial differential equations associated with the Laplace and heat operators, and various generalizations thereof. As a typical reader, we have in mind a student, familiar with the basic concepts of probability based on measure theory, at the level of the graduate texts of Billingsley and Durrett , and who wants a broader perspective on the theory of Brownian motion and related stochastic processes than can be found in these texts.
@techreport {key1802.09679a,
AUTHOR = {Pitman, Jim and Yor, Marc},
TITLE = {A guide to {B}rownian motion stochastic
processes},
TYPE = {preprint},
MONTH = {February},
YEAR = {2018},
NOTE = {ArXiv:1802.09679.},
}
