Brownian motion or Lucretian motion? Which do you prefer?

Why do we call Brownian motion, Brownian motion and not Lucretian motion?

Brownian motion owes its name to the botanist Robert Brown. Brown's famous 1827 experimental observations of pollen grains moving erratically in water are seen as an important step towards the scientific rationalisation of the age old theory of atomisation, attributed to Democritus (ca. 460 BC - ca. 370 BC). The jittery motion of the pollen particles were conjectured to be the result of a perpetual multitude of collisions with tiny invisible particles, Democritus' atoms.

Brown's experiment was, in part, seen as the inspiration for Einstein's 1905 mathematical treatment of what we now call Brownian motion ("On the Motion of Small Particles Suspended in a Stationary Liquid, as Required by the Molecular Kinetic Theory of Heat" Annalen der Physik Volume 322, Issue 8, pages 549-560, 1905). There, Einstein writes "it is possible that the motions described here are identical to the so-called Brownian molecular motion; however, the data available to me are so imprecise that I could not form a judgement on the question".

The naming of this stochastic process after Brown, in light of its importance in formalising Democritus' theory of atomisation, could well have turned out differently had inspiration been sought elsewhere. The six volume poetic work of Titus Lucretius Carus (ca. 99 BC - ca. 55 BC) called "The nature of things" (De rerum natura) presents an atomic-materialist view of the universe in the spirit of Democritus' theory, encouraging the reader to do away with supernatural and divine intervention and to engage with a rational physical explanation of the world around us.

This dread, these shadows of the mind, must thus be swept away
Not by rays of the sun nor by the brilliant beams of day,
But by observing Nature and her laws. And this will lay
The warp out for us - her first principle: that nothing's brought
Forth by any supernatural power out of naught.
For certainly all men are in the clutches of a dread -
Beholding many things take place in heaven overhead
Or here on Earth whose causes they can't fathom, they assign
The explanation for these happenings to powers divine.
Nothing can be made from nothing - once we see that's so,
Already we are on the way to what we want know;
What can things be fashioned from? And how is it, without
The machinations of the gods, all things can come about?

The second volume of Lucretius' work, called "The dance of atoms" attempts to describe what we might call the physics and chemistry of the universe. Therein, we remarkably find the very same intellectual contribution that is so often associated to Robert Brown's pollen experiment.

There's a model, you should realise,
A paradigm of this that's dancing right before your eyes -
For look well when you let the sun peep in a shuttered room
Pouring forth the brilliance of its beams into the gloom,
And you'll see myriads of motes all moving in many ways
Throughout the void and intermingling in the golden rays
As if in everlasting struggle, battling in troops,
Ceaselessly separating and regathering in groups.
From this you can imagine all the motions that take place
Among the atoms that are tossed about in empty space.
For to a certain extent, it is possible for us to trace
Greater things from trivial examples, and discern
In them the train of knowledge. Another reason you should turn
Your attention to the motes that drift and tumble in the light:
Such turmoil means that there are secret motions, out of sight,
That lie concealed in matter. For you'll see the motes careen
Off course, and then bounce back again, by means of blows unseen,
Drifting now in this direction, now that, on every side.
You may be sure this starts with atoms; they are what provide
The base of this unrest. For atoms are moving on their own,
Then small formations of them, nearest them in scale, are thrown
Into agitation by unseen atomic blows,
And these strike slightly larger clusters, and on and on it goes -
A movement that begins on the atomic level, by slight
Degrees ascends until it is perceptible to our sight,
So that we can behold the dust-motes dancing in the sun,
Although the blows that move them can't be seen by anyone.

Depending on who read what and when, "Brownian motion" could well have found itself called "Lucretian motion" (much to the atheist's delight). Perhaps it is not too late.

22/02/12

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A note on branching Lévy processes.
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Martingale Convergence and the Stopped Branching Random Walk.
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Martingale convergence and the functional equation in the multi-type branching random walk. (with A. Rahimzadeh Sani).
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A note on the alpha-quantile option. (with Laura Ballotta).
Applied Mathematical Finance (2001), 8, 137--144.

Perpetual options and Canadization through fluctuation theory. (with Martijn Pistorius).
Annals of Applied Probability (2003), 13(3), 1077-1098.

Law of the iterated logarithm for oscillating random walks conditioned to stay non-negative. (with B. Hambly and G.Kersting)
Stochastic Processes and their Applications (2003), Vol 108/2 pp 327-343.

Some calculations for Israeli options.
Finance and Stochastics. (2004), 8, 73 - 86.

Local extinction versus local exponential growth for spatial branching processes. (with Janos Englander)
Annals of Probability (2004), 32, 78-99.

Exit problems for spectrally negative Lévy processes adn applications to (Canadized) Russian options. (with F.Avram and Martijn Pistorius).
Annals of Applied Probability (2004), 14(1), 215-238.

Travelling wave solutions to the K-P-P equation: alternitives to Simon Harris' probabilistic analysis.
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Ruin probabilties and overshoots for general Lévy insurance risk process. (with C. Kluppelberg and Ross Maller).
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Measure change in multitype branching. (with J.D. Biggins).
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A martingale review of some fluctuation theory for spectrally negative Lévy processes. (with Z. Palmowski).
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Further calculations for Israeli options. (with Erik Baurdoux).
Stochastics. (2004), 76, 549-569.

Asymptotic radial speed of the support of supercritical branching Brownian motion and super-Brownian motion in R^d.
Markov Processes and Related Fields. (2005), 11, 145-156.

Finite expiry Russian options. (with Kees van Schaik and Hans Duistermaat).
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Lévy processes in finance distinguished by their coarse and fine path properties. (with Ronnie Loeffen).
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The smoothing transform: the boundary case. (with J.D. Biggins).
Electronic Journal of Probability (2005), 10, 609-631.

Some remarks on first passage of Lévy process, the American put and pasting principles. (with Larbi Alili).
Annals of Applied Probability (2005), 15, 2062--2080.

On the Novikov-Shiryaev optimal stopping problem in continuous time. (with B. Surya).
Electronic Communications in Probability (2005), 10, 146-154.

Further probabilistic analysis of the Fisher-Kolmogorov-Petrovskii-Piscounov equation: one sided travelling waves. (with John Harris and S. C. Harris).
Annales de l'Instut Henri Poincaré (2006), 42, 125-145.

Overshoots and Undershoots of Lévy process (with Ron Doney).
Annals of Applied Probability (2006), 16, 91-106.

First passage of reflected strictly stable processes.
ALEA (2006), 2, 119-123.

On extreme ruinous behaviour of Lévy insurance risk processes. (with C. Kluppelberg).
Journal of Applied Probability (2006), 43, 594-598.

Quasi-stationary distributions for Lévy process. (with Z. Palmowski).
Bernoulli (2006), 12, 571-581.

Principles of smooth and continuous fit in the determination of endogenous bankruptcy levels. (with B. Surya).
Finance and Stochastics (2007), 11, 131-152.

Pricing Isreali options: a pathwise approach. (with C. Kuhn and K. van Schaik).
Stochastics (2007), 79, 117-137.

Distributional study of De Finetti's dividend problem for a general Lévy insurance risk process. (with Z. Palmowski)
Journal of Applied Probability. (2007), 44, 428-443.

A note on the change of variable formula with local time-space for bounded variation Lévy processes. (With B. Surya).
Séminaire de Probabilité XL. 97-105.

Callable puts as composite exotic options. (with C. Kuhn).
Mathematical Finance (2007), 17, 487-502.

The McKean stochastic game driven by a spectrally negative Lévy process. (with Erik Baurdoux).
Electronic Journal of Probability. (2008), Paper no. 8, 173-197.

Fluctuations of spectrally negative Markov Additive Processes. (with Z. Palmowski).
Séminaire de Probabilité XLI. 121-135.

On the parabolic generator of a general one-dimensional Lévy process. (with Nathalie Eisenbaum).
Electronic Communications in Probability (2008), Paper no. 20, 198-209.

Special, conjugate and complete scale functions for spectrally negative Lévy processes. (with V. Rivero).
Electronic Journal of Probability (2008), Paper no 57, 1672-1701.

Continuous state branching processes and self-similarity. (with Juan Carlos Pardo).
Journal of Applied Probability (2008), 45 (4), 1140-1160.

Analysis of stochastic fluid queues driven by local time processes. (with Takis Konstantopoulos, Paavo Salminen and Marina Sirvio (née Kozlova)).
Advances of Applied Probability (2008), vol. 40 (4), 1072-1103.

Branching processes in random environment die slowly. (with V.Vatutin).
Discrete Mathematics & Theoretical Computer Science Proceedings Fifth Colloquium on Mathematics and Computer Science (2008), 375-396.

Some explicit identities associated with positive self-similar Markov processes. (with Loic Chaumont and Juan Carlos Pardo).
Stochastic Processes and Their Applications (2009), 119/3, 980-1000.

The Shepp-Shiryaev stochastic game driven by a spectrally negative Lévy process. (with Erik Baurdoux).
Theory of Probability and Its Applications (Teoriya Veroyatnostei i ee Primeneniya) (2009), 53, 481-499..

General tax structures and the Lévy insurance risk model. (with Xiaowen Zhou).
Journal of Applied Probability (2009), 46, 1146-1156.

A note on scale functions and the time value of ruin for Lévy insurance risk processes. (with Enrico Biffis).
Insurance Mathematics and Economics (2010), 46, 85-91.

Refracted Lévy processes. (with R. Loeffen).
Annales de l'Instut Henri Poincaré (2010), 46, 24-44.

Strong law of large numbers for fragmentation processes. (with S.C. Harris and R.Knobloch).
Annales de l'Instut Henri Poincaré (2010), 46, 119-134..

Strong law of large numbers for branching diffusions. (with Janos Englander and Simon Harris).
Annales de l'Instut Henri Poincaré (2010), 46, 279-298.

The Wiener-Hopf decomposition.
Encyclopedia of Quantitative Finance, published by Wiley.

Lévy processes.
Encyclopedia of Quantitative Finance, published by Wiley.

Exact and asymptotic n-tuple laws at first and last passage. (with V. Rivero and Juan Carlos Pardo).
Annals of Applied Probability (2010), Vol. 20, No. 2, 522-564.

Convexity and smoothness of scale functions and de Finetti's control problem. (with V. Rivero and Renming Song).
Journal of Theoretical Probability (2010), 23, 547-564.

Old and new examples of scale functions for spectrally negative Lévy processes. (with F. Hubalek).
Sixth Seminar on Stochastic Analysis, Random Fields and Applications, eds R. Dalang, M. Dozzi, F. Russo. Progress in Probability, Birkhäuser (2010), 119-146.

The Gapeev--Kühn stochastic game driven by a spectrally positive Lévy process. (with Erik Baurdoux and Juan Carlos Pardo).
Stochastic Processes and their Applications (2011), 121/6, 1266-1289.

The prolific backbone for supercritical superdiffusions. (with Julien Berestycki and Antonio Murillo).
Stochastic Processes and their Applications (2011), 121/6, 1315-1331.

A Ciesielski-Taylor type identity for positive self-similar Markov processes (with Pierre Patie).
Annales de l'Instut Henri Poincaré (2011), 47/3, 917-928.

Smoothness of scale functions for spectrally negative Lévy processes (with Terence Chan and Mladen Savov).
Probability Theory and Related Fields (2011), 150, 691-708.

On the excursions of reflected local time processes and stochastic fluid queues. (with Takis Konstantopoulos and Paavo Salminen).
Journal of Applied Probability (2011), 48A, 79-98.

Travelling waves and homogeneous fragmentation (with Julien Berestycki and Simon Harris).
Annals of Applied Probability. (2011), 21, 1749-1794.

A Wiener-Hopf Monte Carlo simulation technique for Lévy process (with Alexey Kuznetsov, Juan Carlos Pardo and Kees van Schaik).
Annals of Applied Probability. (2011), 21, 2171-2190.

Backbone decomposition for continuous-state branching processes with immigration. (with Yanxia Ren).
Statistics and Probability Letters. (2012), 82, 139-144.

Fluctuation theory and exit systems for positive self-similar Markov processes (with Loic Chaumont, V. Rivero and Juan Carlos Pardo).
Annals of Probability (2012), 40, 245-279.

Optimal control with absolutely continuous strategies for spectrally negative Lévy processes. (with Ronnie Loeffen and Jose-Luis Pérez).
Journal of Applied Probability (2012), 49, 150-166.

An optimal stopping problem for fragmentation processes. (with Juan Carlos Pardo).
Stochastic Processes and their Applications. (2012), 122, 1210-1225.

Meromorphic Lévy processes and their fluctuation identities (with Alexey Kuznetsov, Juan Carlos Pardo).
Annals of Applied Probability. (2012), 22, 1101-1135.

Supercritical super-Brownian motion with a general branching mechanism and travelling waves (with Antonio Murillo-Salas, Rongli Liu and Yanxia Ren).
Annales de l'Instut Henri Poincaré. (2012), 48, 661-687.

Super-Brownian motion: Lp-convergence of martingales through the pathwise spine decomposition. (with Antonio Murillo-Salas).
In Advances in Superprocesses and Nonlinear PDEs Series: Springer Proceedings in Mathematics & Statistics, Vol. 38 Englander, Janos; Rider, Brian C. (Eds.), 2013.

An application of the backbone decomposition to supercritical super-Brownian motion with a barrier. (with Antonio Murillo-Salas and Jose-Luis Pérez).
Journal of Applied Probability. (2012), 49, 671-684.

Spectrally negative Lévy processes perturbed by functionals of their running supremum (with Curdin Ott).
Journal of Applied Probability (2012), 49, 1005-1014.

On optimal dividends in the dual model (with Erhan Bayraktar and Kazutoshi Yamazaki).
ASTIN Bulletin (2013), 43, 359-372.

Pricing of Contingent Convertibles under Smile Conform Models. (with José Manuel Corcuera, Jan de Spiegeleer, Albert Ferreiro-Castilla, Dillip Madan and Wim Schoutens).
Journal of Credit Risk (2013), 9(3), 121-140.

Multilevel Monte Carlo simulation for Lévy processes based on the Wiener-Hopf factorisation (with Albert Ferreiro-Castilla, Rob Scheichl and Gowri Suryanarayana).
Stochastic Processes and its Applications (2014), 124, 985-1010.

Optimal dividends in the dual model under transaction costs (with Erhan Bayraktar and Kazutoshi Yamazaki).
Insurance: Mathematics and Economics (2014), 54, 133-143.

Hitting distributions of alpha-stable processes via path censoring and self-similarity (with Juan Carlos Pardo and Alex Watson).
Annals of Probability. (2014), 42, 398-430.

A capped optimal stopping problem for the maximum process (with Curdin Ott).
Acta Applicandae Mathematicae. (2014), 129, 147-174.

The total mass of super-Brownian motion upon exiting balls and Sheu's compact support condition. (with Marion Hesse).
Stochastic Processes and its Applications (2014), 124, 2003-2022.

The hitting time of zero for a stable process (with Alexey Kuznetsov, Juan Carlos Pardo and Alex Watson).
Electronic Journal of Probability (2014), 19, 1-26.

Survival of homogenous fragmentation processes with killing. (with Robert Knobloch).
Annales de l'Instut Henri Poincaré(2014), 50, 476-491.

New families of subordinators with explicit transition probability semigroup. (with James Burridge, Mateusz Kwasnicki, Alexey Kuznetsov.).
Stochastic Processes and their Applications (2014), 124, 3480-3495.

Occupation times of refracted Lévy processes (with Juan Carlos Pardo and Jose-Luis Pérez).
Journal of Theoretical Probability. (2014), 27, 1292-1315.

The extended hypergeometric class of Lévy processes (with Juan Carlos Pardo and Alex Watson).
Journal of Applied Probability. (2014), 51A, 391-408.

The backbone decomposition for spatially dependent supercritical superprocesses (with Jose-Luis Pérez and Yanxia Ren).
Séminaire de Probabilité XLVI. (2015), 33-60.

Potentials of stable processes (with Alex Watson).
Séminaire de Probabilité XLVI. (2015), 333-344.

Spines, skeletons and the Strong Law of Large Numbers for superdiffusions (with Maren Eckhoff and Matthias Winkel).
Annals of Probability. (2015), Vol. 43, No. 5, 2594-2659.

Branching Brownian motion in a strip: survival near criticality. (with Simon Harris and Marion Hesse).
Annals of Probability. (2016), Vol. 44, No. 1, 235-275.

An Euler-Poisson Scheme for Lévy driven SDEs (with Albert Ferreiro-Castilla and Rob Scheichl).
Journal of Applied Probability. (2016), 53, 262-278.

UK universities find a cash cow in the financial fall-out
The Conversation. (See also the longer version: The UK financial mathematics M.Sc.)

Deep factorisation of the stable process.
Electronic Journal of Probability. (2016), Volume 21 paper no. 23, 1-28..

More on hypergeometric Lévy processes (with Emma Horton).
Advances in Applied Probability (2016), 48A, 153 - 158.

Optimal prediction for positive self-similar Markov processes (With Erik Baurdoux and Curdin Ott).
Electronic Journal of Probability. (2016), Volume 21 paper no. 48, 1-24.

Perpetual integrals for Lévy processes (with Leif Döring).
Journal of Theoretical Probability (2016), Volume 29, Issue 3, pp 1192-1198.

The largest fragment of a homogeneous fragmentation process (With Peter Mörters and Francis Lane).
Journal of Statistical Physics. (2017), 166, 1226-1246.

Real self-similar processes started from the origin (with Steffen Deriech and Leif Döring).
Annals of Probability. (2017), Volume 45, No. 3, 1952-2003.

Conditioning subordinators embedded in Markov processes. (with Victor Rivero, and Bati Sengul ).
Stochastic Processes and their Applications. (2017), 127, 1234-1254.

A phase transition in excursions from infinity of the "fast" fragmentation-coalescence process. (with Steven Pagett, Tim Rogers and Jason Schweinsberg).
Annals of Probability. (2017), Volume 45, No. 6A, 3829-3849.

Deep factorisation of the stable process II: potentials and applications. (with Victor M. Rivero and Bati Sengul).
Annales de l'Instut Henri Poincaré. (2018), Vol. 54, No. 1, 343-362.

Universality in a class of fragmentation-coalescence processes. (with Steven Pagett and Tim Rogers).
Annales de l'Instut Henri Poincaré. (2018), Vol. 54, No. 2, 1134-1151.

Almost sure growth of supercritical multi-type continuous state branching process (With Sandra Palau and Yanxia Ren).
ALEA, Lat. Am. J. Probab. Math. Stat. (2018), 15, 409-428

Stable processes, self-similarity and the unit ball
ALEA, Lat. Am. J. Probab. Math. Stat. (2018), 15, 617-690.

Unbiased "walk-on-spheres" Monte Carlo methods for the fractional Laplacian (With Ana Osojnik and Tony Shardlow).
IMA J. Numer. Anal. (2018), 38 no. 3, 1550-1578.

Stable windings at the origin. (With Stavros Vakeroudis).
Stochastic Processes and their Applications (2018), 128, 4309-4325.

Extinction properties of multi-type continuous-state branching processes (With Sandra Palau).
Stochastic Processes and their Applications (2018), 128 3466–3489.

Conditioned real self-similar Markov processes. (with Victor M. Rivero and Weerapat Satitkanitkul).
Stochastic Processes and their Applications (2019), 129, 954-977.

Multi-species neutron transport equation. (with Alex M.G. Cox, Simon C. Harris and Emma Horton).
Journal of Statistical Physics (2019), 176(2), 425-455.

Skeletal stochastic differential equations for continuous-state branching process (With Dorottya Fekete and Joaquin Fontbona).
Journal of Applied Probability (2019), 56, 1122-1150.

Entrance and exit at infinity for stable jump diffusions (with Leif Döring).
Annals of Probability (2020), Vol. 48, No. 3, 1220-1265

Entrance laws at the origin of self-similar Markov processes in high dimensions. (with Victor M. Rivero, Bati Sengul and Ting Yang ) .
Transactions of the American Mathematical Society (2020) Vol. 373 (9), 6227-6299

Stable processes conditioned to avoid an interval (with Leif Döring and Philip Weissmann).
Stochastic Processes and their Applications (2020), Vol 130 no. 2, 471-487.

Skeletal stochastic differential equations for superprocesses (With Dorottya Fekete and Joaquin Fontbona).
Journal of Applied Probability. (2020) 57, 1111-1134.

Deep factorisation of the stable process III: radial excursion theory and the point of closest reach. (with Victor M. Rivero and Weerapat Satitkanitkul).
Potential Analysis (2020), 53, 1347-1375.

Stochastic Methods for the Neutron Transport Equation I: Linear Semigroup asymptotics. (with Emma Horton and Denis Villemonais) .
Annals of Applied Probability (2020) Vol. 30, No. 6, 2815-2845.

Stochastic Methods for the Neutron Transport Equation II: Almost sure growth. (with Emma Horton and Simon Harris) .
Annals of Applied Probability (2020), Vol. 30, No. 6, 2573-2612.

Stochastic Methods for the Neutron Transport Equation III: Generational many-to-one and k_eff. (with Alex M.G. Cox, Emma Horton and Denis Villemonais) .
SIAM Journal of Applied Mathematics (2021), 81(3), 982–1001.

Attraction to and repulsion from a subset of the unit sphere for isotropic stable Lévy processes (With Sandra Palau, and Tsogzolmaa Saizmaa).
Stochastic Processes and their Applications (2021), 137 272-293.

Double hypergeometric Lévy processes and self-similarity (With Juan Carlos Pardo and Matija Vidmar).
Journal of Applied Probability (2021), 58, 254–273.

Stable Lévy processes in a cone. (with Victor M. Rivero and Weerapat Satitkanitkul).
Annales de l'Instut Henri Poincaré (2021), 57, No. 4, 2066–2099.

A Lifetime of Excursions Through Random Walks and Lévy Processes (With Loic Chaumont).
In: A Lifetime of Excursions Through Random Walks and Lévy Processes: A Volume in Honour of Ron Doney’s 80th Birthday. (2022) p1-11. Birkhäuser 2022.

A Transformation for Spectrally Negative Lévy Processes and Applications (With Marie Chazal and Pierre Patie).
In: A Lifetime of Excursions Through Random Walks and Lévy Processes: A Volume in Honour of Ron Doney’s 80th Birthday. (2022) p157-180. Birkhäuser 2022.

The Doob–McKean identity for stable Lévy processes (With Neil O'Connell).
In: A Lifetime of Excursions Through Random Walks and Lévy Processes: A Volume in Honour of Ron Doney’s 80th Birthday. p269-282. Birkhäuser 2022.

Oscillatory attraction and repulsion from a subset of the unit sphere or hyperplane for isotropic stable Lévy processes (With Mateusz Kwaśniki, Sandra Palau, and Tsogzolmaa Saizmaa).
In: A Lifetime of Excursions Through Random Walks and Lévy Processes: A Volume in Honour of Ron Doney’s 80th Birthday. p283-313, Birkhäuser 2022.

An optimal stopping problem for spectrally negative Markov additive processes (With Mine Çağlar, and Ceren Vardar-Acar).
Stochastic Processes and their Applications (2022), 150, 1109-1138

Yaglom limit for non-local critical branching Markov processes (With Emma Horton , Simon Harris and Minmin Wang) .
Annals of Probability (2022), Vol. 50, No. 6, 2373–2408.

General path integrals and stable SDEs (with Sam Baguley and Leif Döring).
To appear in the Journal of the European Mathematical Society

Monte-Carlo Methods for the Neutron Transport Equation (with Alex M.G. Cox, Simon Harris and Minmin Wang) .
To appear in SIAM Journal of Uncertainty Quantification.

Asymptotic moments of spatial branching processes (with Emma Horton and Isaac Gonzalez).
To appear in Probability Theory and Related Fields. (open access https://doi.org/10.1007/s00440-022-01131-2). See also the Erratum for the non-critical cases.

Multi-type Λ-coalescents (with Samuel Johnston and Tim Rogers).
To appear in Annals of Applied Probability.