Theory and Applications of Stochastic Processes: An Analytical Approach

486 Pages·2010·8.84 MB·English

by  Zeev Schuss (auth.)

Applied Mathematical Sciences
Volume 170
Editors
S.S. Antman J.E. Marsden L. Sirovich
Advisors
J.K. Hale P. Holmes J. Keener
J. Keller B.J. Matkowsky A. Mielke
C.S. Peskin K.R. Sreenivasan
Forfurthervolumes:
http://www.springer.com/series/34
Zeev Schuss
Theory and Applications of
Stochastic Processes
An Analytical Approach
Zeev Schuss
Department of Applied Mathematics
School of Mathematical Science
Tel AAviv University
69 978 Tel Aviv
Israel
Editors:
S.S.Antman J.E.Marsden L.Sirovich
DepartmentofMathematics ControlandDynamical LaboratoryofApplied
and Systems,107-81 Mathematics
InstituteforPhysicalScience CaliforniaInstitute Departmentof
andTechnology ofTechnology BiomathematicalSciences
UniversityofMaryland Pasadena,CA91125 MountSinaiSchoolofMedicine
CollegePark USA NewYork,NY10029-6574
MD20742-4015 [email protected] USA

USA [email protected]
[email protected]
ISSN 0 066-5452
ISBN 978-1-4419-1604-4 e-ISBN 978-1-4419-1605-1
DOI 10.1007/978-1-4419-1605-1
Springer New York Dordrecht He idelberg London

Library of Congress Control Number: 2009942425

Mathematics Subject Classification (2000): 60-01, 60J60, 60J65, 60J70, 60J75, 60J50, 60J35, 60J05,
60J25, 60J27, 58J37, 37H10, 60H10, 65C30, 60G40, 46N55, 47N55, 60F05, 60F10
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Preface
Stochastic processes and diffusion theory are the mathematical underpinnings of
manyscientificdisciplines,includingstatisticalphysics,physicalchemistry,molec-
ularbiophysics,communicationstheory,andmanymore.Manybooks,reviews,and
researcharticleshavebeenpublishedonthistopic,fromthepurelymathematicalto
the most practical. Some are listed below in alphabetical order, but a Web search
revealsmanymore.
• Mathematicaltheoryofstochasticprocesses[14],[25],[46],[47],[53],[58],
[57], [72], [76], [74], [82], [206], [42], [101], [106], [115], [116], [117],
[150],[153],[161],[208],[234],[241]
• Stochasticdynamics[5],[80],[84],[171],[193],[204],[213]
• Numerical analysis of stochastic differential equations [207], [59], [132],
[131],[103],[174],[206],[4]
• Largedeviationstheory[32],[52],[54],[68],[77],[110],[55]
• Statisticalphysics[38],[82],[100],[98],[99],[185],[206],[196],[242]
• Electricalengineering[108],[148],[199]
This book offers an analytical approach to stochastic processes that are most
commoninthephysicalandlifesciences. Itsaimistomakeprobabilitytheoryin
function space readily accessible to scientists trained in the traditional methods of
applied mathematics, such as integral, ordinary, and partial differential equations
andasymptoticmethods,ratherthaninprobabilityandmeasuretheory.
Thebookpresentsthebasicnotionsofstochasticprocesses, mostlydiffusions,
including fundamental concepts from measure theory in the space of continuous
functions(thespaceofrandomtrajectories). TherudimentsoftheWienermeasure
infunctionspaceareintroducednotforthesakeofmathematicalrigor,butratheras
atoolforcountingtrajectories,whicharetheelementaryeventsintheprobabilistic
description of experiments. The Wiener measure turns out to be appropriate for
the statistical description of nonequilibrium systems and replaces the equilibrium
Boltzmanndistributioninconfigurationspace.Therelevantstochasticprocessesare
mainlystochasticdifferentialequationsdrivenbyEinstein’swhiteortheOrnstein–
Uhlenbeck (OU) colored noise, but include also continuous-time Markovian jump
v
vi Preface
processesanddiscrete-timesemi-Markovianprocesses,suchasrenewalprocesses.
Continuous-ordiscrete-timejumpprocessesoftencontainmoreinformationabout
the origin of randomness in a given system than diffusion (white noise) models.
Approximationsofthemoremicroscopicdiscretemodelsbytheircontinuumlimits
(diffusions)arethecornerstoneofmodelingrandomsystems.
The analytical approach relies heavily on initial and boundary value problems
for partial differential equations that describe important probabilistic quantities,
suchasprobabilitydensityfunctions,meanfirstpassagetimes,densityofthemean
timespentatapoint,survivalprobability,probabilityfluxdensity,andsoon.Green’s
functionanditsderivativesplayacentralroleinexpressingthesequantitiesanalyti-
callyandindeterminingtheirinterrelationships. Themostprominentequationsare
the Fokker–Planck and Kolmogorov equations for the transition probability den-
sity, which are the well-known transport-diffusion partial differential equations of
continuumtheory, theKolmogorovandFeynman–Kacrepresentationformulasfor
solutions of initial boundary value problems in terms of conditional probabilities,
theAndronov–Vitt–Pontryaginequationfortheconditionalmeanfirstpassagetime,
andmore.
Computersimulationsarethenumericalrealizationsofstochasticdynamicsand
areubiquitousincomputationalphysics,chemistry,molecularbiophysics,andcom-
municationstheory.Thebehaviorofrandomtrajectoriesnearboundariesofthesim-
ulation impose a variety of boundary conditions on the probability density and its
functionals. Thequiteintricateconnectionbetweentherandomtrajectoriesandthe
boundaryconditionsforthepartialdifferentialequationsistreatedherewithspecial
care. TheWienerpathintegralandWienermeasureinfunctionspaceareusedex-
tensivelyfordeterminingtheseconnections.Someofthesimulation-orientedtopics
are discussed in one dimension only, because the book is not about simulations.
SpecialtopicsthatariseinBrowniandynamicssimulationsandtheiranalysis,such
assimulatingtrajectoriesbetweenconstantconcentrations,connectingasimulation
tothecontinuum,thealbedoproblem,behavioroftrajectoriesathigher-dimensional
partiallyreflectingboundaries,theestimateofrareeventsduetoescapeofrandom
trajectories through small openings, the analysis of simulations of interacting par-
ticles and more are discussed in [214]. The important topic of nonlinear optimal
filtering theory and performance evaluation of optimal phase trackers is discussed
in[215].
Theextractionofusefulinformationaboutstochasticprocessesrequirestheso-
lution of the basic equations of the theory. Numerical solutions of integral and
partialdifferentialequationsorcomputersimulationsofstochasticdynamicsareof-
teninadequatefortheexplorationoftheparameterspaceorforstudyingrareevents.
Themethodsandtricksofthetradeofappliedmathematicsareusedextensivelyto
obtainexplicitanalyticalapproximationstothesolutionsofcomplicatedequations.
ThemethodsincludesingularperturbationtechniquessuchastheWKBexpansion,
borrowed from quantum mechanics, boundary layer theory, borrowed from con-
tinuum theory, the Wiener–Hopf method, and more. These methods are the main
analyticaltoolsfortheanalysisofsystemsdrivenbysmallnoise(smallrelativeto
thedrift).Inparticular,theyprovideanalyticalexpressionsforthemeanexittimeof
Preface vii
arandomtrajectoryfromagivendomain,aproblemoftenencounteredinphysics,
physical chemistry, molecular and cellular biology, and in other applications. The
last chapter, on stochastic stability, illustrates the power of the analytical methods
forestablishingthestochasticstability,orevenforstabilizingunstablestructuresby
noise.
The book contains exercises and worked-out examples. The hands-on training
in stochastic processes, as my long teaching experience shows, consists of solv-
ingtheexercises,withoutwhichunderstandingisonlyillusory. Thebookisbased
on lecture notes from a one- or two-semester course on stochastic processes and
their applications that I taught many times to graduate students of mathematics,
appliedmathematics,physics,chemistry,computerscience,electricalengineering,
and other disciplines. My experience shows that mathematical rigor and applica-
tionscannotcoexistinthesamecourse;excessiverigorleavesnoroomforin-depth
developmentofmodelingmethodologyandturnsoffstudentsinterestedinscientific
andengineeringapplications. Thereforethebookcontainsonlytheminimalmathe-
maticalrigorrequiredforunderstandingthenecessarymeasure-theoreticalconcepts
andforenablingthestudentstousetheirownjudgmentofwhatiscorrectandwhat
requiresfurthertheoreticalstudy.Thestudentshouldbefamiliarwithbasicmeasure
andintegrationtheoryaswellaswithrudimentsoffunctionalanalysis,astaughtin
mostgraduateanalysiscourses(see,e.g.,[79]). Moreintricatemeasure-theoretical
problemsarediscussedintheoreticaltexts,asmentionedinthefirstlistofreferences
above. Thefirstcoursecancoverthefundamentalconceptspresentedinthefirstsix
chaptersandeitherChapter8onMarkovprocessesandtheirdiffusionapproxima-
tions,orChapters7or11,whichdevelopsomeofthementionedmethodsofapplied
mathematicsandapplythemtothestudyofsystemsdrivenbysmallnoise.Chapters
7and9–12,whichdevelopandapplytheanalyticaltoolstothestudyoftheequa-
tions developed in Chapters 1–6 and 8, can be chosen for the second semester. A
well-pacedcoursecancovermostofthebookinasinglesemester. Thetwobooks
[214], onBrowniandynamicsandsimulations, and[215], onoptimalfilteringand
phasetracking,canbeusedforone-semesterspecialtopicsfollowupcourses.
It is recommended that students of Chapters 7 and 9–12 acquire some inde-
pendenttraininginsingularperturbationmethods,forexample,fromclassicaltexts
suchas[194],[121],[20],[195],[251]. Thesechaptersrequirethebasicknowledge
ofChapters1–6andasolidtraininginpartialdifferentialequationsofmathematical
physicsandintheasymptoticmethodsofappliedmathematics.
viii Preface
Acknowledgment. Muchofthematerialpresentedinthisbookisbasedonmycol-
laborationwithmanyscientistsandstudents,whosenamesarelistednexttominein
theauthorindex. Muchoftheworkdoneinthenearly30yearssincemyfirstbook
appeared [213], was initiated in the Department of Applied Mathematics and En-
gineeringScienceatNorthwesternUniversityandintheDepartmentofMolecular
BiophysicsandPhysiologyatRushMedicalCenterinChicago,forwhosehospital-
ityIamgrateful. ThescientificenvironmentprovidedbyTel-AvivUniversity, my
homeinstitution,wasconducivetointerdisciplinarycooperation. Itsdouble-major
undergraduate program in mathematics and physics, as well as its M.Sc. program
in E.E. produced my most brilliant graduate students and collaborators in applied
mathematics.
ZeevSchuss
List of Figures
2.1 FourBrowniantrajectoriessampledatdiscretetimes. Threecylin-
ders are marked by vertical bars. The trajectories were sampled
accordingtothescheme(2.21). . . . . . . . . . . . . . . . . . . . . 29
2.2 ThreeBrowniantrajectoriessampledatdiscretetimesaccordingto
theWienermeasurePr {·}bythescheme(2.21). . . . . . . . . . . 41
0
2.3 The graphs of X (t) (dots), its first refinement X (t) (dot-dash),
1 2
andsecondrefinementX (t)(dash). . . . . . . . . . . . . . . . . . 50
3
2.4 ABrowniantrajectorysampledat1024points. . . . . . . . . . . . 50
6.1 ThepotentialU(x)=x2/2−x4/4(solidline)andtheforceF(x)=
−U0(x)(dashedline). Thedomainofattractionofthestableequi-
libriumx=0istheintervalD =(−1,1). . . . . . . . . . . . . . . 186
6.2 The solution (6.31) of the noiseless dynamics (6.30) with x(0) =
0.99 inside (bottom line) and x(0) = 1.01 outside (top line) the
domainofattractionD. . . . . . . . . . . . . . . . . . . . . . . . . 187
8.1 The potential U(x) = 3(x−1)2 −2(x−1)3. The well is x <
x = 2, the bottom is at x = x = 1, the top of the barrier is
C A
C =(x ,U(x )),andtheheightofthebarrierisE =U(x )−
C C C C
U(x )=1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
A
8.2 The energy contours y2/2+U(x) = E for the potential U(x) =
x2/2−x3/3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
8.3 ThewashboardpotentialU(x)=cosx−IxforI =0.3. . . . . . . 286
8.4 The stable running solution (top) and separatrices, 2π apart in the
phasespace(x,y),forγ =0.1andI =0.3. Thethickcurveisthe
criticaltrajectoryS =S thattouchesthex-axis.Theenergyofthe
C
system on the running solution slides to −∞ down the washboard
potentialinFigure8.3. . . . . . . . . . . . . . . . . . . . . . . . . 287
8.5 In the limit γ → ∞ the stable running solution of Figure 8.4 dis-
appearstoinfinityandthecriticalenergycontourS = S andthe
C
separatricescoalesce. . . . . . . . . . . . . . . . . . . . . . . . . . 288
8.6 TheeffectiveglobalpotentialΦ(A)(8.131)inthesteady-stateFPE
(8.130). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
ix