@article{pitman1999laplace,
  abstract = {Various known expressions in terms of hyperbolic functions for the Laplace transforms of random times related to one-dimensional Brownian motion are derived in a unified way by excursion theory and extended to one-dimensional diffusions.},
  added-at = {2010-08-26T21:00:05.000+0200},
  author = {Pitman, Jim and Yor, Marc},
  biburl = {http://www.bibsonomy.org/bibtex/232487a00421279724f13507d75b3e2d1/peter.ralph},
  copyright = {Copyright © 1999 International Statistical Institute (ISI) and Bernoulli Society for Mathematical Statistics and Probability},
  interhash = {98b83794f95cf988701997710f33a078},
  intrahash = {32487a00421279724f13507d75b3e2d1},
  issn = {13507265},
  journal = {Bernoulli},
  jstor_articletype = {research-article},
  jstor_formatteddate = {Apr., 1999},
  keywords = {diffusions excursions hitting_times laplace_transform},
  language = {English},
  number = 2,
  pages = {pp. 249-255},
  publisher = {International Statistical Institute (ISI) and Bernoulli Society for Mathematical Statistics and Probability},
  timestamp = {2010-08-26T21:00:05.000+0200},
  title = {Laplace Transforms Related to Excursions of a One-Dimensional Diffusion},
  url = {http://www.jstor.org/stable/3318434},
  volume = 5,
  year = 1999
}

@article{pitman2003hitting,
  abstract = {{Summary: Basic relations between the distributions of hitting, occupation and inverse local times of a one-dimensional diffusion process $X$, first discussed by {\it K. It\^o} and {\it H. P. McKean jun.} [``Diffusion processes and their sample paths'' (1965; Zbl 0127.09503)], 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 $T$.}},
  added-at = {2010-08-26T20:57:36.000+0200},
  author = {Pitman, Jim and Yor, Marc},
  biburl = {http://www.bibsonomy.org/bibtex/2af18ef33b84971e852358ff3c9728f8d/peter.ralph},
  classmath = {{*60J55 (Additive functionals) }},
  doi = {10.3150/bj/1068129008},
  interhash = {39ddb848ac7a374fa90954b46177cbd8},
  intrahash = {af18ef33b84971e852358ff3c9728f8d},
  journal = {Bernoulli },
  keywords = {diffusions excursions hitting_times laplace_transform local_times occupation_time review},
  language = {English},
  number = 1,
  pages = {1-24},
  timestamp = {2010-08-26T20:57:36.000+0200},
  title = {Hitting, occupation and inverse local times of one-dimensional diffusions: {Martingale} and excursion approaches.},
  volume = 9,
  year = 2003
}

@article{permanpitmanyor92,
  abstract = {ome 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.},
  added-at = {2010-01-15T19:31:30.000+0100},
  author = {Perman, M. and Pitman, J. and Yor, M.},
  biburl = {http://www.bibsonomy.org/bibtex/2c36b8cb697027e4d7261bae24c95a168/peter.ralph},
  interhash = {995d0b64de56aeabd2e0fa51cf4c508c},
  intrahash = {c36b8cb697027e4d7261bae24c95a168},
  journal = {Probability Theory and Related Fields},
  keywords = {Poisson_process excursions size_biased_parititions},
  number = 1,
  pages = {21--39},
  publisher = {Springer},
  timestamp = {2010-01-15T19:31:30.000+0100},
  title = {{Size-biased sampling of Poisson point processes and excursions}},
  url = {http://www.springerlink.com/content/n08754882308h67n/},
  volume = 92,
  year = 1992
}

@article{pitman2003pkp,
  abstract = {    This paper presents some general formulas for random partitions of a finite set
derived by Kingman's model of random sampling from an interval partition generated
by subintervals whose lengths are the points of a Poisson point process. These lengths
can be also interpreted as the jumps of a subordinator, that is an increasing process
with stationary independent increments. Examples include the two-parameter family
of Poisson-Dirichlet models derived from the Poisson process of jumps of a stable sub-
ordinator. Applications are made to the random partition generated by the lengths
of excursions of a Brownian motion or Brownian bridge conditioned on its local time
at zero.
},
  added-at = {2009-04-25T01:55:19.000+0200},
  author = {Pitman, J.},
  biburl = {http://www.bibsonomy.org/bibtex/2868b2332af933348e30042a963ace490/peter.ralph},
  description = {{Poisson-Kingman partitions}},
  interhash = {a630b147e8314ea6d732b75770f2fd90},
  intrahash = {868b2332af933348e30042a963ace490},
  journal = {Lecture Notes-Monograph Series},
  keywords = {Poisson_Dirichlet_distribution exchangeability excursions partitions reference sampling_formula},
  pages = {1--34},
  publisher = {Institute of Mathematical Statistics},
  timestamp = {2009-04-25T01:55:19.000+0200},
  title = {{Poisson-Kingman partitions}},
  url = {http://arxiv.org/pdf/math/0210396},
  year = 2003
}

@article{kuchler1,
  added-at = {2009-04-24T23:33:01.000+0200},
  author = {K{\"u}chler, Uwe},
  biburl = {http://www.bibsonomy.org/bibtex/207931e3aac0bf06022d5984877d60b04/peter.ralph},
  coden = {KRMPBV},
  comments = {tangential.},
  description = {q-paper},
  fjournal = {Kyoto University. Research Institute for Mathematical
               Sciences. Publications},
  interhash = {9d2960fb68cada351dcf10f0da53e12d},
  intrahash = {07931e3aac0bf06022d5984877d60b04},
  issn = {0034-5318},
  journal = {Publ. Res. Inst. Math. Sci.},
  keywords = {excursions quasidiffusions spectral_theory},
  mrclass = {60J65},
  mrnumber = {MR574035 (81i:60075)},
  mrreviewer = {Kazuaki Taira},
  number = 1,
  pages = {245--268},
  timestamp = {2009-04-24T23:33:01.000+0200},
  title = {Some asymptotic properties of the transition densities of
               one-dimensional quasidiffusions},
  volume = 16,
  year = 1980
}

@article{kuchler3,
  added-at = {2009-04-24T23:33:01.000+0200},
  author = {K{\"u}chler, Uwe},
  biburl = {http://www.bibsonomy.org/bibtex/28c52b2b13279e8d54c02b0fb4c02f9a9/peter.ralph},
  comments = {tangential.},
  description = {q-paper},
  fjournal = {Mathematische Operationsforschung und Statistik Series
               Statistics},
  interhash = {cee186157e3787b34e4f4aa8fd049c17},
  intrahash = {8c52b2b13279e8d54c02b0fb4c02f9a9},
  issn = {0323-3944},
  journal = {Math. Operationsforsch. Statist. Ser. Statist.},
  keywords = {excursions quasidiffusions spectral_theory},
  mrclass = {60J25 (62M05)},
  mrnumber = {MR658237 (84d:60108b)},
  mrreviewer = {No{\"e}l Veraverbeke},
  number = 2,
  pages = {219--230},
  timestamp = {2009-04-24T23:33:01.000+0200},
  title = {Exponential families of {M}arkov processes. {II}.
               {B}irth-and-death processes},
  volume = 13,
  year = 1982
}

@article{kuchler2,
  added-at = {2009-04-24T23:33:01.000+0200},
  author = {K{\"u}chler, Uwe},
  biburl = {http://www.bibsonomy.org/bibtex/2415dbbd8efc0c62092b8f79a8f2693b6/peter.ralph},
  comments = {tangential.},
  description = {q-paper},
  fjournal = {Mathematische Operationsforschung und Statistik Series
               Statistics},
  interhash = {5a7dc59bb31b1503d4df545eb3f20023},
  intrahash = {415dbbd8efc0c62092b8f79a8f2693b6},
  issn = {0323-3944},
  journal = {Math. Operationsforsch. Statist. Ser. Statist.},
  keywords = {excursions quasidiffusions spectral_theory},
  mrclass = {60J25 (62M05)},
  mrnumber = {MR665063 (84d:60108a)},
  mrreviewer = {No{\"e}l Veraverbeke},
  number = 1,
  pages = {57--69},
  timestamp = {2009-04-24T23:33:01.000+0200},
  title = {Exponential families of {M}arkov processes. {I}. {G}eneral
               results},
  volume = 13,
  year = 1982
}

@incollection{kuchler4,
  abstract = { condensed version of "On sojourn times..." },
  added-at = {2009-04-24T23:33:01.000+0200},
  address = {Utrecht},
  author = {K{\"u}chler, Uwe},
  biburl = {http://www.bibsonomy.org/bibtex/27eafb45253187674f7b43acd62344cb1/peter.ralph},
  booktitle = {Probability theory and mathematical statistics, Vol.\ II
               (Vilnius, 1985)},
  description = {q-paper},
  interhash = {175786bf1f5e32f259791a7971db70cc},
  intrahash = {7eafb45253187674f7b43acd62344cb1},
  keywords = {excursions quasidiffusions spectral_theory},
  mrclass = {60J55 (60J60)},
  mrnumber = {MR901531 (88j:60130)},
  pages = {161--165},
  publisher = {VNU Sci. Press},
  timestamp = {2009-04-24T23:33:01.000+0200},
  title = {On {I}t\^o's excursion law, local times and spectral measures for quasidiffusions},
  year = 1987
}

@incollection{kuchler5,
  added-at = {2009-04-24T23:33:01.000+0200},
  address = {Berlin},
  author = {K{\"u}chler, Uwe},
  biburl = {http://www.bibsonomy.org/bibtex/2534e34f8c9a769b48b2218948eed041d/peter.ralph},
  booktitle = {Markov processes and control theory (Gau\ss ig, 1988)},
  comments = {tangential.},
  description = {q-paper},
  interhash = {82e682ba16ff25eef1b6a534663e76c0},
  intrahash = {534e34f8c9a769b48b2218948eed041d},
  keywords = {excursions quasidiffusions spectral_theory},
  mrclass = {60J60},
  mrnumber = {MR1075621 (91m:60146)},
  pages = {100--103},
  publisher = {Akademie-Verlag},
  series = {Math. Res.},
  timestamp = {2009-04-24T23:33:01.000+0200},
  title = {A limit theorem for the excursion of quasidiffusions straddling {$t$}},
  volume = 54,
  year = 1989
}

@article{kuchler7,
  added-at = {2009-04-24T23:33:01.000+0200},
  author = {K{\"u}chler, Uwe},
  biburl = {http://www.bibsonomy.org/bibtex/2a63480d9c07006befd87872623bbc116/peter.ralph},
  coden = {CRABAA},
  comments = {More about sojourn times being mixtures of exponentials.},
  description = {q-paper},
  fjournal = {Doklady Bolgarsko\u\i\ Akademii Nauk. Comptes Rendus de
              l'Acad\'emie Bulgare des Sciences},
  interhash = {2e4e6524d2bb0e3e04a92eaf0e5945c6},
  intrahash = {a63480d9c07006befd87872623bbc116},
  issn = {0366-8681},
  journal = {C. R. Acad. Bulgare Sci.},
  keywords = {excursions spectral_theory},
  mrclass = {60J25},
  mrnumber = {MR825106 (87f:60111)},
  mrreviewer = {Stig I. Rosenlund},
  number = 11,
  pages = {1445--1448},
  timestamp = {2009-04-24T23:33:01.000+0200},
  title = {Quasidiffusions, sojourn times and spectral measures},
  volume = 38,
  year = 1985
}

@article{kuchler6,
  added-at = {2009-04-24T23:33:01.000+0200},
  author = {K{\"u}chler, Uwe},
  biburl = {http://www.bibsonomy.org/bibtex/283f87ce5fb90f6885ae7b0ccfab06446/peter.ralph},
  coden = {JMKYAZ},
  comments = {Shows that sojourn times are mixtures of exponentials, related to the
  		"dual string", and more similar results.},
  description = {q-paper},
  fjournal = {Journal of Mathematics of Kyoto University},
  interhash = {0eff0525aeb3a63b9609d1357c425743},
  intrahash = {83f87ce5fb90f6885ae7b0ccfab06446},
  issn = {0023-608X},
  journal = {J. Math. Kyoto Univ.},
  keywords = {excursions quasidiffusions spectral_theory},
  mrclass = {60J55 (60J60)},
  mrnumber = {MR857226 (87m:60173)},
  mrreviewer = {John Kent},
  number = 3,
  pages = {403--421},
  timestamp = {2009-04-24T23:33:01.000+0200},
  title = {On sojourn times, excursions and spectral measures connected with quasidiffusions},
  volume = 26,
  year = 1986
}

@article{pitman-yor-97,
  abstract = {The two-parameter Poisson-Dirichlet distribution, denoted $\mathsf{PD}(\alpha, \theta)$ 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 $\mathsf{PD}(0, \theta)$. Known properties of $\mathsf{PD}(0, \theta)$, including the Markov chain description due to Vershik, Shmidt and Ignatov, are generalized to the two-parameter case. The size-biased random permutation of $\mathsf{PD}(\alpha, \theta)$ 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 $0 < \alpha < 1, \mathsf{PD}(\alpha, 0)$ 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 $\mathsf{PD}(1/2, 0)$, and the corresponding distribution for a Brownian bredge is $\mathsf{PD}(1/2, 1/2)$. The $\mathsf{PD}(\alpha, 0)$ and $\mathsf{PD}(\alpha, \alpha)$ 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 $\alpha$.},
  added-at = {2009-04-01T01:40:08.000+0200},
  author = {Pitman, J. and Yor, M.},
  biburl = {http://www.bibsonomy.org/bibtex/2a46c28a46dc7cd2c51757d95eb542856/peter.ralph},
  interhash = {57e97b344c4678b523121d4fb1bd57ed},
  intrahash = {a46c28a46dc7cd2c51757d95eb542856},
  journal = {Annals of Probability},
  keywords = {GEM_distribution Poisson_Dirichlet_distribution coalescent_theory excursions partitions subordinators urn_models},
  number = 2,
  pages = {855--900},
  publisher = {Hayward, Calif.[etc.] Institute of Mathematical Statistics.},
  timestamp = {2009-04-01T01:40:08.000+0200},
  title = {{The two-parameter Poisson-Dirichlet distribution derived from a stable subordinator}},
  url = {http://stat-www.berkeley.edu/users/pitman/433.pdf},
  volume = 25,
  year = 1997
}

@article{Scheffer1995,
  added-at = {2008-04-22T14:25:45.000+0200},
  author = {Scheffer, Carel L.},
  biburl = {http://www.bibsonomy.org/bibtex/29d9bd81822f18fb8d04375eea4c9b8c7/smicha},
  description = {Stochastic Processes and their Applications},
  interhash = {69b97d27ead42bc99e94b8683aa81d03},
  intrahash = {9d9bd81822f18fb8d04375eea4c9b8c7},
  journal = {Stochastic Processes and their Applications},
  keywords = {Excursions},
  month = Jan,
  number = 1,
  pages = {101--118},
  timestamp = {2008-04-22T14:25:45.000+0200},
  title = {The rank of the present excursion},
  url = {http://www.sciencedirect.com/science/article/B6V1B-3YYTDND-7/1/e53566fbaa08038bfcc71264e30eb7ab},
  volume = 55,
  year = 1995
}

@incollection{statphys23_1079,
  abstract = {In the 1960s Mandelbrot developed the use of fractals to  describe
how the shape of many aspects of the natural  world departs from the
Euclidean. In particular he  proposed two kinds of fractal model to
capture the way in  which natural data is often persistent in time
(his Joseph effect, common in hydrology and exemplified by 
fractional Brownian motion) and or prone to heavy tailed  jumps (the Noah
effect, typical of economic index time  series, for  
which he gave L\'{e}vy flights as an exemplar). Both effects are  now 
well demonstrated in proxies both for the Earth's auroral electric
currents and  for the turbulent solar wind which is their ultimate energy
source. Modelling, however, has usually emphasised  one  of the
Noah and Joseph parameters (the tail exponent $\mu$ and one derived
from the temporal behaviour such as power spectral $\beta$) at the other's
expense. This poster will first describe recent work [1] in which we applied a simple self-affine stable model-linear fractional stable motion, LFSM, which unifies both  effects-to give insight into space physics data. I will show how we have resolved some contradictions seen in earlier work, where purely Joseph or Noah descriptions  had been sought. Such hybrid Noah-Joseph ambivalent [2] behaviour is  highly
topical in physics. It is typically studied in the  paradigm of the
continuous time random walk (CTRW) rather  than LFSM. Intriguingly the
self-similarity exponent  extracted from the CTRW differs from that
seen in LFSM,  being a ratio of $\mu$ and a temporal exponent  rather 
than an additive function. The poster will elucidate the physical  
differences between these two pictures with reference to a newly-derived diffusion equation for LFSM, which replaces the second order spatial derivative in the equation of fBm [3] with a fractional derivative of order $\mu$.
I will also show work in progress  using  an LFSM generator and simple 
analytic scaling arguments to study the problem of the  area between a
fractional L\'{e}vy curve and a threshold-related both to Bernoulli excursions 
and to the burst size measure introduced  by Takalo and Consolini into solar-terrestrial physics and further studied by  Freeman et al [4,5].
Finally I will discuss how LFSM gives the appearance of multi-affine scaling without having an underlying turbulent cascade or other multiplicative process. The importance of this property for the  interpretation of natural time series will be discussed.

1) Watkins et al, Space Sci. Rev. 121, 271, 2005.\\
2) Brockmann et al, Nature 439, 462, 2006.\\
3) Wang and Lung, Phys. Lett. A 151, 119, 1990.\\
4) Freeman et al, Geophys. Res. Lett. 27, 1367, 2000.\\
5) Freeman et al, Phys. Rev. E 62, 8794, 2000.},
  added-at = {2007-06-20T10:16:09.000+0200},
  address = {Genova, Italy},
  author = {Watkins, N.W. and Credgington, D. and Sanchez, R. and Chapman, S.C.},
  biburl = {http://www.bibsonomy.org/bibtex/29f4390d7d0965c4c7371b02ddd8f6c7b/statphys23},
  booktitle = {Abstract Book of the XXIII IUPAP International Conference on Statistical Physics},
  editor = {Pietronero, Luciano and Loreto, Vittorio and Zapperi, Stefano},
  interhash = {0e87d0cbad594262f413cf438663a492},
  intrahash = {9f4390d7d0965c4c7371b02ddd8f6c7b},
  keywords = {anomalous bernoulli diffusion excursions fractional linear motion stable statphys23 topic-3},
  month = {9-13 July},
  timestamp = {2007-06-20T10:16:09.000+0200},
  title = {A diffusion equation for linear fractional stable motion, apparent multifractality  \& applications to space physics},
  url = {http://st23.statphys23.org/webservices/abstract/preview_pop.php?ID_PAPER=1079},
  year = 2007
}