Research ArticleModeling Anomalous Diffusion by a SubordinatedIntegrated Brownian Motion
Long Shi12 and Aiguo Xiao1
1School of Mathematics and Computational Science Xiangtan University Xiangtan Hunan 411105 China2School of Science Central South University of Forest and Technology Changsha Hunan 410004 China
Correspondence should be addressed to Long Shi slong8008163com
Received 6 February 2017 Accepted 23 March 2017 Published 4 April 2017
Academic Editor Giorgio Kaniadakis
Copyright copy 2017 Long Shi and Aiguo Xiao This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
We consider a particular type of continuous time random walk where the jump lengths between subsequent waiting times arecorrelated In a continuum limit the process can be defined by an integrated Brownian motion subordinated by an inverse 120572-stablesubordinator We compute the mean square displacement of the proposed process and show that the process exhibits subdiffusionwhen 0 lt 120572 lt 13 normal diffusion when 120572 = 13 and superdiffusion when 13 lt 120572 lt 1 The time-averaged mean squaredisplacement is also employed to show weak ergodicity breaking occurring in the proposed process An extension to the fractionalcase is also considered
1 Introduction
Anomalous diffusion is found in a wide diversity of systems(see review articles [1ndash4] and references therein) In onedimension it is characterized by a mean square displacement(MSD) of the form
with 120572 = 1 which deviates from the linear dependenceon time found in normal diffusion The coefficient 119870120572 isgeneralized diffusion constant It is called subdiffusion for0 lt 120572 lt 1 and superdiffusion for 120572 gt 1 [2]
A fundamental account to anomalous diffusion is pro-vided by a stochastic process called continuous time randomwalk (CTRW) which was originally introduced by Montrolland Weiss in 1965 [5] In a continuum limit the processhas been considered by Fogedby [6] via coupled Langevinequations
where 120585(119904) is a white Gaussian noise with ⟨120585(119904)⟩ = 0⟨120585(119904)120585(1199041015840)⟩ = 120575(119904 minus 1199041015840) and 120577(119904) is a white 120572-stable Levy noisetaking positive values only and independent of 120585(119904)
In (2) the randomwalk 119909(119905) is parametrized in terms of acontinuous variable 119904 which is subjected to a random timechange This random time change to the physical time 119905 isdescribed by the second equation The combined process inthe physical time is then given by 119909(119905) = 119883(119904(119905)) where 119904(119905)is the inverse process to 119905(119904) defined as
Mathematically the fundamental approach to describethe combined process 119909(119905) = 119883(119904(119905)) is based on subordi-nation technique which was first introduced by Bochner [7]Using the notation of subordination the process 119883(119904) 119905(119904)and 119904(119905) are named parent process subordinator and inversesubordinator respectively
In recent years (2) consisting of Brownian motions withor without external field and inverse 120572-stable subordinatorare becoming a hot topic [8ndash17] There are also severalother processes considered as parent processes within thesubordination framework for example Levy-stable process
HindawiAdvances in Mathematical PhysicsVolume 2017 Article ID 7246865 7 pageshttpsdoiorg10115520177246865
2 Advances in Mathematical Physics
[18 19] arithmetic Brownian motion [20] geometric Brow-nian motion [21 22] Ornstein-Uhlenbeck process [23 24]tempered stable process [25] fractional Brownian motion[26 27] and fractional Levy-stable process [28] Here wenote that apart from inverse 120572-stable subordinator inversetempered 120572-stable subordinator and infinitely divisible sub-ordinators are also considered in the literatures [16 20 25ndash31]
In the simplest CTRW process after each jumps a newpair of waiting time and jump length is drawn from theassociated distributions independent of the previous valuesThis independence giving rise to a renewal process is notalways justified for instance by observations of humanmotion patterns [32] and active biological movements [33] orin financial market dynamics [34] Recently three correlatedCTRW models are introduced to model the random walkswith some forms of memory [35ndash37] Some advances inthe field of CTRWs with correlated temporal orand spatialstructure can be also found in [38ndash45]
In this work we consider a jump-correlated CTRWmodel which has the subordination form 119909(119905) = 119883(119904120572(119905))Here the parent process 119883(119904) is an integrated Brownianmotion defined by
The integrated Brownian motion 119883(119904) is called the ran-dom acceleration process in the physical literature and hasbeen studied by many authors For instance it appears in thecontinuum description of the equilibrium Boltzmann weightof a semiflexible polymer chain [46] It also appears in thedescription of statistical properties of the Burgers equationwith Brownian initial velocity [47] Some further results ofthe integrated Brownian motion can be found in the paper[48] reviewing this subject
The structure of the paper is as follows In Section 2 weintroduce the jump-correlated CTRW model In Section 3we compute MSD of the proposed process and observethe corresponding anomalous diffusive behaviors The time-averaged MSD is also employed to show weak ergodicitybreaking occurring in the proposed process In Section 4 wegeneralize the integrated Brownian motion to the fractionalintegral of Brownianmotion and compute the correspondingMSDThe conclusions are given in Section 5
2 Model
We begin by recalling the general framework for CTRW the-ory Let 119879119894119894ge1 be the sequence of nonnegative independentidentically distributed (IID) random variable representingwaiting times between jumps of a particle We set 119905(0) = 0and 119905(119899) = sum119899119894=1 119879119894 that is the time of the 119899th jump Let 119869119894119894ge1be the sequence of IID jump lengths of the particle which areassumed to be independent of waiting timesWe set119883(0) = 0
and 119883(119899) = sum119899119894=1 119869119894 that is the position of the particle afterthe 119899th jump Then the position of the particle at time 119905 isgiven by
where 119873(119905) = max119899 ge 0 119905(119899) le 119905 is the number of jumpsup to time 119905 The process 119909(119905) = 119883(119873(119905)) is called CTRW
In what follows we analyze a particular type of CTRWwhere the jump lengths are correlated Assume that eachjump is equal to
where 120585119895 are IID randomvariables with finite secondmoment(for simplicity we assume that their second moment isequal to 1) Moreover we assume that each waiting time119879119894 is nonnegative IID random variable whose characteristicfunction 120593(119896) is given by
120593 (119896) = exp minus |119896|120572 exp (minus1198941205871205722 sgn (119896))
0 lt 120572 le 1(8)
In the continuous limit we get the following set ofcoupled Langevin equations for the position 119909 and time 119905 ofthe CTRW
where 120585(119904) and 120577(119904) are the same as those in (2) and 119861(119904) isthe standard Brownian motion with ⟨119861(119904)⟩ = 0 ⟨119861(119904)119861(119905)⟩ =min(119904 119905)
An equivalent representation of (9) in the form ofsubordination is
where 119905(119904) = int1199040120577(1199041015840)1198891199041015840 is an 120572-stable totally skewed Levy
motion with characteristic function
⟨119890minus119906119905(119904)⟩ = exp minus119906120572119904 0 lt 120572 le 1 (13)
3 Discussions
At first let us compute the MSD of subordinated process119909(119905) = 119883(119904120572(119905))
Advances in Mathematical Physics 3
Assume that 119901(119909 119905) 119891(119909 119904) and 119892(119904 119905) are PDFs ofsubordinated process 119909(119905) parent process 119883(119904) and inversesubordinator 119904120572(119905) respectively In terms of subordinationwe have
It is easy to observe from (25) that the process issubdiffusive when 0 lt 120572 lt 13 normally diffusive when120572 = 13 and superdiffusive when 13 lt 120572 le 1
It is well-known that the MSD of the process given by (2)is of the form
Comparing (25) with (26) we see that Fogedbyrsquos model canonly represent anomalous subdiffusion but our model canrepresent subdiffusion normal diffusion and superdiffusion
Next we study weak ergodicity breaking of the subordi-nated process 119909(119905)
In an ergodic system one can find the equivalence
⟨(Δ119909)2⟩ (Δ) = ⟨1205752 (Δ)⟩ (27)
Here 1205752(Δ) is the time-averaged MSD of the process 119909(119905)defined as
1205752 (Δ) = 1119879 minus Δ int119879minusΔ
0[119909 (119905 + Δ) minus 119909 (119905)]2 119889119905 (28)
where Δ is the lag time and 119879 is the overall measure timeFor anomalous diffusion the behavior of the ensemble
MSD ⟨(Δ119909)2⟩(Δ) and the time-averaged MSD (28) may befundamentally different The disparity ⟨(Δ119909)2⟩(Δ) = ⟨1205752(Δ)⟩is usually calledweak ergodicity breaking (orweak nonergod-icity) [49] In recent years weak nonergodicity of anomalousdiffusion process attracts more and more attentions [49ndash55]
Since for any 119886 gt 0 parent process119883(119904) satisfies119883 (119886119904) = int119886119904
comparing (33) with (34) we see that the linear lag timedependence of ⟨1205752(Δ)⟩ is different from the power-law formΔ3120572 of ⟨(Δ119909)2⟩(Δ) which implies that subordinated process119909(119905) is weakly nonergodic
At last we consider the propagator 119901(119909 119905) associatedwith the subordinated process 119909(119905) By the total probabilityformula we obtain an integral representation of 119901(119909 119905)
where 0119868119901119905 is the Riemann-Liouville fractional integrationoperator of order 119901 defined by [56]
0119868119901119905 119891 (119905) = 1Γ (119901) int
119905
0(119905 minus 120591)119901minus1 119891 (120591) 119889120591 (119901 gt 0) (45)
As a result the jump-correlated CTRW has the subordi-nation form 119909(119905) = 119883(119904120572(119905)) where parent process 119883(119904) is ofthe form (44) and inverse subordinator 119904120572(119905) is defined by(12)
Here we are interested in the competition between thememory parameter 120583 and stability index 120572 In what followswe will not discuss any properties of motion other than theMSD
In terms of (44) we get
⟨1198832 (119904)⟩ = 1Γ2 (1 minus 120583)
sdot int1199040
1198891199041015840(119904 minus 1199041015840)120583 int
119904
0
⟨119861 (1199041015840) 119861 (11990410158401015840)⟩(119904 minus 11990410158401015840)120583 11988911990410158401015840
= 1Γ2 (1 minus 120583) int
119904
0
1198891199041015840(119904 minus 1199041015840)120583 int
119904
0
min 1199041015840 11990410158401015840(119904 minus 11990410158401015840)120583 11988911990410158401015840
= 1Γ2 (1 minus 120583) int
119904
0
1198891199041015840(119904 minus 1199041015840)120583 [int
1199041015840
0
11990410158401015840(119904 minus 11990410158401015840)120583 119889119904
10158401015840
+ int1199041199041015840
1199041015840(119904 minus 11990410158401015840)120583 119889119904
10158401015840]
(46)
By denoting 119868(119904) = int1199040(1198891199041015840(119904 minus 1199041015840)120583) int1199041015840
0(11990410158401015840(119904 minus 11990410158401015840)120583)11988911990410158401015840 and
exchanging the order of quadratic integral 119868(119904) we obtain119868 (119904) = int119904
0
11988911990410158401015840(119904 minus 11990410158401015840)120583 int
119904
11990410158401015840
11990410158401015840(119904 minus 1199041015840)120583 119889119904
1015840 (47)
Thus
⟨1198832 (119904)⟩= 2
Γ2 (1 minus 120583) int119904
0
1199041015840(119904 minus 1199041015840)120583 119889119904
1015840 int1199041199041015840
1(119904 minus 11990410158401015840)120583 119889119904
10158401015840
= 2(1 minus 120583) Γ2 (1 minus 120583) int
119904
01199041015840 (119904 minus 1199041015840)1minus2120583 1198891199041015840
where 119870120583 = 2119861(2 2 minus 2120583)(1 minus 120583)Γ2(1 minus 120583) and 119861(119886 119887) =int10119909119886minus1(1 minus 119909)119887minus1119889119909 is Beta functionWe observe from (48) that in the limiting case 120583 rarr 0
memory function119872(119905) = 1 the parent process 119883(119904) definedby (43) reduces to the form defined by (11) and the secondmoment of 119883(119904) computed by (48) reduces to 11987001199043 where1198700 = 2119861(2 2) = 13 the same form as (16) We are alsointerested in the limiting case 120583 rarr 1 At the moment the
memory function 119872(119905) is a Dirac 120575-function 119883(119904) definedby (43) reduces to the standard Brownian motion
Let us turn to the MSD of the subordinated process 119909(119905)In terms of (18) and (48) we obtain
= 119870120583 Γ (4 minus 2120583)119906120572(3minus2120583)+1
(50)
Taking the inverse Laplace transform for ⟨(Δ119909)2⟩(119906) we have⟨(Δ119909)2⟩ (119905) = 119870120583120572119905120572(3minus2120583) 0 lt 120583 lt 1 0 lt 120572 le 1 (51)
where 119870120583120572 = 119870120583Γ(4 minus 2120583)Γ(120572(3 minus 2120583) + 1) In the limitingcase 120583 rarr 0 the parameter119870120583120572 reduces to
Thus (51) reduces to (25)It is easy to observe from (51) that there exists a compe-
tition between the memory parameter 120583 and stability index120572 For the case 120572 le 13 the subordinated process exhibitssubdiffusive behaviors For the case 13 lt 120572 lt 1 the processis subdiffusive when 1 lt 3minus2120583 lt 1120572 normal diffusive when3 minus 2120583 = 1120572 and superdiffusive when 1120572 lt 3 minus 2120583 lt 35 Conclusions
We introduce an integrated Brownian motion subordinatedby inverse 120572-stable one-sided Levy motion which is acontinuous limit of a jump-correlated CTRW In terms of theensembleMSD of the proposed process we conclude that theprocess is subdiffusive when 0 lt 120572 lt 13 normal diffusivewhen 120572 = 13 and superdiffusive when 13 lt 120572 le 1 Thetime-averagedMSD is also employed to showweak ergodicitybreaking occurring in the proposed process
We also generalize the process to the case where theparent process is fractional integral of Brownian motion Interms of the MSD we observe a competition between thememory parameter 120583 and stability index 120572 Other types ofinverse subordinators may be also candidates
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (Grant no 11671343) and the ScientificResearch Project of Hunan Provincial EducationDepartment(no 17B258)
6 Advances in Mathematical Physics
References
[1] J-P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990
[2] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[3] R Metzler and J Klafter ldquoThe restaurant at the end of therandomwalk recent developments in the description of anoma-lous transport by fractional dynamicsrdquo Journal of Physics AMathematical and General vol 37 no 31 pp R161ndashR208 2004
[4] I Eliazar and J Klafter ldquoAnomalous is ubiquitousrdquo Annals ofPhysics vol 326 no 9 pp 2517ndash2531 2011
[5] E W Montroll and G H Weiss ldquoRandom walks on lattices IIrdquoJournal of Mathematical Physics vol 6 pp 167ndash181 1965
[6] H C Fogedby ldquoLangevin equations for continuous time Levyflightsrdquo Physical Review E vol 50 no 2 pp 1657ndash1660 1994
[7] S Bochner ldquoDiffusion equation and stochastic processesrdquoProceedings of the National Academy of Sciences vol 35 no 7pp 368ndash370 1949
[8] A Baule and R Friedrich ldquoJoint probability distributions for aclass of non-Markovian processesrdquo Physical Review E vol 71no 2 Article ID 026101 2005
[9] A Piryatinska A I Saichev and W A Woyczynski ldquoModelsof anomalous diffusion the subdiffusive caserdquo Physica AStatistical Mechanics and Its Applications vol 349 no 3-4 pp375ndash420 2005
[10] M Magdziarz A Weron and K Weron ldquoFractional Fokker-Planck dynamics stochastic representation and computer sim-ulationrdquo Physical Review EmdashStatistical Nonlinear and SoftMatter Physics vol 75 no 1 Article ID 016708 2007
[11] D Kleinhans and R Friedrich ldquoContinuous-time randomwalks simulation of continuous trajectoriesrdquo Physical Review Evol 76 no 6 Article ID 061102 2007
[12] AWeronMMagdziarz and KWeron ldquoModeling of subdiffu-sion in space-time-dependent force fields beyond the fractionalFokker-Planck equationrdquo Physical Review E vol 77 no 3Article ID 036704 2008
[13] M Magdziarz A Weron and J Klafter ldquoEquivalence of thefractional fokker-planck and subordinated langevin equationsthe case of a time-dependent forcerdquo Physical Review Letters vol101 no 21 Article ID 210601 2008
[14] A Weron and S Orzel ldquoIto formula for subordinated Langevinequation A case of time dependent forcerdquoActa Physica PolonicaB vol 40 no 5 pp 1271ndash1277 2009
[15] S Eule and R Friedrich ldquoSubordinated Langevin equationsfor anomalous diffusion in external potentialsmdashbiasing anddecoupled external forcesrdquo EPL vol 86 no 3 Article ID 300082009
[16] M Magdziarz ldquoLangevin picture of subdiffusion with infinitelydivisiblewaiting timesrdquo Journal of Statistical Physics vol 135 no4 pp 763ndash772 2009
[17] M Magdziarz ldquoStochastic representation of subdiffusion pro-cesses with time-dependent driftrdquo Stochastic Processes andTheirApplications vol 119 no 10 pp 3238ndash3252 2009
[18] M Magdziarz and A Weron ldquoCompetition between subdiffu-sion and Levy flights a Monte Carlo approachrdquo Physical ReviewE vol 75 Article ID 056702 2007
[19] B o Dybiec and E Gudowska-Nowak ldquoSubordinated diffusionand continuous time random walk asymptoticsrdquo Chaos vol 20no 4 Article ID 043129 2010
[20] A Wyłomanska ldquoArithmetic Brownian motion subordinatedby tempered stable and inverse tempered stable processesrdquoPhysica A StatisticalMechanics and Its Applications vol 391 no22 pp 5685ndash5696 2012
[21] H Gu J-R Liang and Y-X Zhang ldquoOn a time-changedgeometric Brownian motion and its application in financialmarketrdquo Acta Physica Polonica B vol 43 no 8 pp 1667ndash16812012
[22] J Gajda and A Wyłomanska ldquoGeometric Brownian motionwith tempered stable waiting timesrdquo Journal of StatisticalPhysics vol 148 no 2 pp 296ndash305 2012
[23] J Janczura S Orzeł and A Wyłomanska ldquoSubordinated 120572120572-stable Ornstein-Uhlenbeck process as a tool for financial datadescriptionrdquo Physica A Statistical Mechanics and its Applica-tions vol 390 no 23-24 pp 4379ndash4387 2011
[24] J Gajda and AWyłomanska ldquoTime-changedOrnsteinndashUhlen-beck processrdquo Journal of Physics A Mathematical and Theoreti-cal vol 48 no 13 Article ID 135004 2015
[25] A Wyłomanska ldquoThe tempered stable process with infinitelydivisible inverse subordinatorsrdquo Journal of StatisticalMechanicsTheory and Experiment vol 2013 no 10 Article ID P10011 2013
[26] Y-X Zhang H Gu and J-R Liang ldquoFokker-planck typeequations associated with subordinated processes controlled bytempered 120572-stable processesrdquo Journal of Statistical Physics vol152 no 4 pp 742ndash752 2013
[27] J Gajda and A Wyłomanska ldquoFokkerndashPlanck type equationsassociated with fractional Brownian motion controlled byinfinitely divisible processesrdquo Physica A Statistical Mechanicsand Its Applications vol 405 pp 104ndash113 2014
[28] M Teuerle A Wyłomanska and G Sikora ldquoModeling anoma-lous diffusion by a subordinated fractional Levy-stable processrdquoJournal of Statistical Mechanics Theory and Experiment vol2013 no 5 Article ID P05016 2013
[29] J Gajda andMMagdziarz ldquoFractional Fokker-Planck equationwith tempered 120572-stable waiting times Langevin picture andcomputer simulationrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 011117 2010
[30] J Janczura and A Wyłomanska ldquoAnomalous diffusion mod-els different types of subordinator distributionrdquo Acta PhysicaPolonica B vol 43 no 5 pp 1001ndash1016 2012
[31] J Gajda ldquoFractional FokkerndashPlanck equationwith space depen-dent drift and diffusion the case of tempered 120572-stable waiting-timesrdquo Jagellonian University Institute of Physics Acta PhysicaPolonica B vol 44 no 5 pp 1149ndash1161 2013
[32] C Song T Koren P Wang and A Barabasi ldquoModelling thescaling properties of humanmobilityrdquoNature Physics vol 6 no10 pp 818ndash823 2010
[33] A Maye C-H Hsieh G Sugihara and B Brembs ldquoOrder inspontaneous behaviorrdquo PLoS ONE vol 2 no 5 article e4432007
[34] E Scalas ldquoThe application of continuous-time randomwalks infinance and economicsrdquo Physica A Statistical Mechanics and itsApplications vol 362 no 2 pp 225ndash239 2006
[35] A Chechkin M Hofmann and I M Sokolov ldquoContinuous-time random walk with correlated waiting timesrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 80no 3 Article ID 031112 2009
[36] M M Meerschaert E Nane and Y Xiao ldquoCorrelated continu-ous time random walksrdquo Statistics amp Probability Letters vol 79no 9 pp 1194ndash1202 2009
Advances in Mathematical Physics 7
[37] V Tejedor and R Metzler ldquoAnomalous diffusion in correlatedcontinuous time random walksrdquo Journal of Physics A Mathe-matical and Theoretical vol 43 no 8 Article ID 082002 2010
[38] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E vol 85 no 5 Article ID 051103 2012
[39] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time randomwalksmdashscaling limits andLangevin picturerdquo Journal of Statistical Mechanics Theory andExperiment vol 2012 no 4 Article ID P04010 2012
[40] M Magdziarz W Szczotka and P Zebrowski ldquoAsymptoticbehaviour of randomwalks with correlated temporal structurerdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 469 no 2159 Article ID 201304192013
[41] J H Schulz A V Chechkin and R Metzler ldquoCorrelatedcontinuous time random walks combining scale-invariancewith long-range memory for spatial and temporal dynamicsrdquoJournal of Physics A Mathematical and Theoretical vol 46 no47 Article ID 475001 2013
[42] L Shi Z Yu H Huang ZMao andA Xiao ldquoThe subordinatedprocesses controlled by a family of subordinators and corre-sponding FokkerndashPlanck type equationsrdquo Journal of StatisticalMechanics Theory and Experiment vol 2014 no 12 Article IDP12002 2014
[43] J Wang J Zhou L Lv W Qiu and F Ren ldquoHeterogeneousmemorized continuous time random walks in an external forcefieldsrdquo Journal of Statistical Physics vol 156 no 6 pp 1111ndash11242014
[44] F-Y Ren J Wang L-J Lv H Pan and W-Y Qiu ldquoEffect ofdifferent waiting time processes with memory to anomalousdiffusion dynamics in an external force fieldsrdquo Physica AStatistical Mechanics and Its Applications vol 417 pp 202ndash2142015
[45] L Lv F-Y Ren J Wang and J Xiao ldquoCorrelated continuoustime random walk with time averaged waiting timerdquo Physica AStatistical Mechanics and Its Applications vol 422 pp 101ndash1062015
[46] T W Burkhardt ldquoSemiflexible polymer in the half plane andstatistics of the integral of a Brownian curverdquo Journal of PhysicsA Mathematical and General vol 26 no 22 pp L1157ndashL11621993
[47] P Valageas ldquoStatistical properties of the burgers equation withbrownian initial velocityrdquo Journal of Statistical Physics vol 134no 3 pp 589ndash640 2009
[48] TW Burkhardt ldquoThe random acceleration process in boundedgeometriesrdquo Journal of Statistical Mechanics Theory and Exper-iment vol 2007 no 7 Article ID P07004 2007
[49] J Jeon A V Chechkin and R Metzler ldquoScaled Brownianmotion a paradoxical process with a time dependent diffusivityfor the description of anomalous diffusionrdquo Physical ChemistryChemical Physics vol 16 no 30 pp 15811ndash15817 2014
[50] Y He S Burov R Metzler and E Barkai ldquoRandom time-scaleinvariant diffusion and transport coefficientsrdquo Physical ReviewLetters vol 101 no 5 Article ID 058101 2008
[51] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 2009
[52] J-H Jeon and R Metzler ldquoFractional Brownian motion andmotion governed by the fractional Langevin equation in con-fined geometriesrdquo Physical Review E vol 81 no 2 Article ID021103 2010
[53] E Barkai Y Garini and R Metzler ldquoStrange kinetics of singlemolecules in living cellsrdquo Physics Today vol 65 no 8 pp 29ndash352012
[54] J Kursawe J Schulz and R Metzler ldquoTransient aging infractional Brownian and Langevin-equation motionrdquo PhysicalReview E vol 88 no 6 Article ID 062124 2013
[55] R Metzler J-H Jeon A G Cherstvy and E Barkai ldquoAnoma-lous diffusion models and their properties non-stationaritynon-ergodicity and ageing at the centenary of single particletrackingrdquo Physical Chemistry Chemical Physics vol 16 no 44pp 24128ndash24164 2014
[56] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[18 19] arithmetic Brownian motion [20] geometric Brow-nian motion [21 22] Ornstein-Uhlenbeck process [23 24]tempered stable process [25] fractional Brownian motion[26 27] and fractional Levy-stable process [28] Here wenote that apart from inverse 120572-stable subordinator inversetempered 120572-stable subordinator and infinitely divisible sub-ordinators are also considered in the literatures [16 20 25ndash31]
In the simplest CTRW process after each jumps a newpair of waiting time and jump length is drawn from theassociated distributions independent of the previous valuesThis independence giving rise to a renewal process is notalways justified for instance by observations of humanmotion patterns [32] and active biological movements [33] orin financial market dynamics [34] Recently three correlatedCTRW models are introduced to model the random walkswith some forms of memory [35ndash37] Some advances inthe field of CTRWs with correlated temporal orand spatialstructure can be also found in [38ndash45]
In this work we consider a jump-correlated CTRWmodel which has the subordination form 119909(119905) = 119883(119904120572(119905))Here the parent process 119883(119904) is an integrated Brownianmotion defined by
The integrated Brownian motion 119883(119904) is called the ran-dom acceleration process in the physical literature and hasbeen studied by many authors For instance it appears in thecontinuum description of the equilibrium Boltzmann weightof a semiflexible polymer chain [46] It also appears in thedescription of statistical properties of the Burgers equationwith Brownian initial velocity [47] Some further results ofthe integrated Brownian motion can be found in the paper[48] reviewing this subject
The structure of the paper is as follows In Section 2 weintroduce the jump-correlated CTRW model In Section 3we compute MSD of the proposed process and observethe corresponding anomalous diffusive behaviors The time-averaged MSD is also employed to show weak ergodicitybreaking occurring in the proposed process In Section 4 wegeneralize the integrated Brownian motion to the fractionalintegral of Brownianmotion and compute the correspondingMSDThe conclusions are given in Section 5
2 Model
We begin by recalling the general framework for CTRW the-ory Let 119879119894119894ge1 be the sequence of nonnegative independentidentically distributed (IID) random variable representingwaiting times between jumps of a particle We set 119905(0) = 0and 119905(119899) = sum119899119894=1 119879119894 that is the time of the 119899th jump Let 119869119894119894ge1be the sequence of IID jump lengths of the particle which areassumed to be independent of waiting timesWe set119883(0) = 0
and 119883(119899) = sum119899119894=1 119869119894 that is the position of the particle afterthe 119899th jump Then the position of the particle at time 119905 isgiven by
where 119873(119905) = max119899 ge 0 119905(119899) le 119905 is the number of jumpsup to time 119905 The process 119909(119905) = 119883(119873(119905)) is called CTRW
In what follows we analyze a particular type of CTRWwhere the jump lengths are correlated Assume that eachjump is equal to
where 120585119895 are IID randomvariables with finite secondmoment(for simplicity we assume that their second moment isequal to 1) Moreover we assume that each waiting time119879119894 is nonnegative IID random variable whose characteristicfunction 120593(119896) is given by
120593 (119896) = exp minus |119896|120572 exp (minus1198941205871205722 sgn (119896))
0 lt 120572 le 1(8)
In the continuous limit we get the following set ofcoupled Langevin equations for the position 119909 and time 119905 ofthe CTRW
where 120585(119904) and 120577(119904) are the same as those in (2) and 119861(119904) isthe standard Brownian motion with ⟨119861(119904)⟩ = 0 ⟨119861(119904)119861(119905)⟩ =min(119904 119905)
An equivalent representation of (9) in the form ofsubordination is
where 119905(119904) = int1199040120577(1199041015840)1198891199041015840 is an 120572-stable totally skewed Levy
motion with characteristic function
⟨119890minus119906119905(119904)⟩ = exp minus119906120572119904 0 lt 120572 le 1 (13)
3 Discussions
At first let us compute the MSD of subordinated process119909(119905) = 119883(119904120572(119905))
Advances in Mathematical Physics 3
Assume that 119901(119909 119905) 119891(119909 119904) and 119892(119904 119905) are PDFs ofsubordinated process 119909(119905) parent process 119883(119904) and inversesubordinator 119904120572(119905) respectively In terms of subordinationwe have
It is easy to observe from (25) that the process issubdiffusive when 0 lt 120572 lt 13 normally diffusive when120572 = 13 and superdiffusive when 13 lt 120572 le 1
It is well-known that the MSD of the process given by (2)is of the form
Comparing (25) with (26) we see that Fogedbyrsquos model canonly represent anomalous subdiffusion but our model canrepresent subdiffusion normal diffusion and superdiffusion
Next we study weak ergodicity breaking of the subordi-nated process 119909(119905)
In an ergodic system one can find the equivalence
⟨(Δ119909)2⟩ (Δ) = ⟨1205752 (Δ)⟩ (27)
Here 1205752(Δ) is the time-averaged MSD of the process 119909(119905)defined as
1205752 (Δ) = 1119879 minus Δ int119879minusΔ
0[119909 (119905 + Δ) minus 119909 (119905)]2 119889119905 (28)
where Δ is the lag time and 119879 is the overall measure timeFor anomalous diffusion the behavior of the ensemble
MSD ⟨(Δ119909)2⟩(Δ) and the time-averaged MSD (28) may befundamentally different The disparity ⟨(Δ119909)2⟩(Δ) = ⟨1205752(Δ)⟩is usually calledweak ergodicity breaking (orweak nonergod-icity) [49] In recent years weak nonergodicity of anomalousdiffusion process attracts more and more attentions [49ndash55]
Since for any 119886 gt 0 parent process119883(119904) satisfies119883 (119886119904) = int119886119904
comparing (33) with (34) we see that the linear lag timedependence of ⟨1205752(Δ)⟩ is different from the power-law formΔ3120572 of ⟨(Δ119909)2⟩(Δ) which implies that subordinated process119909(119905) is weakly nonergodic
At last we consider the propagator 119901(119909 119905) associatedwith the subordinated process 119909(119905) By the total probabilityformula we obtain an integral representation of 119901(119909 119905)
where 0119868119901119905 is the Riemann-Liouville fractional integrationoperator of order 119901 defined by [56]
0119868119901119905 119891 (119905) = 1Γ (119901) int
119905
0(119905 minus 120591)119901minus1 119891 (120591) 119889120591 (119901 gt 0) (45)
As a result the jump-correlated CTRW has the subordi-nation form 119909(119905) = 119883(119904120572(119905)) where parent process 119883(119904) is ofthe form (44) and inverse subordinator 119904120572(119905) is defined by(12)
Here we are interested in the competition between thememory parameter 120583 and stability index 120572 In what followswe will not discuss any properties of motion other than theMSD
In terms of (44) we get
⟨1198832 (119904)⟩ = 1Γ2 (1 minus 120583)
sdot int1199040
1198891199041015840(119904 minus 1199041015840)120583 int
119904
0
⟨119861 (1199041015840) 119861 (11990410158401015840)⟩(119904 minus 11990410158401015840)120583 11988911990410158401015840
= 1Γ2 (1 minus 120583) int
119904
0
1198891199041015840(119904 minus 1199041015840)120583 int
119904
0
min 1199041015840 11990410158401015840(119904 minus 11990410158401015840)120583 11988911990410158401015840
= 1Γ2 (1 minus 120583) int
119904
0
1198891199041015840(119904 minus 1199041015840)120583 [int
1199041015840
0
11990410158401015840(119904 minus 11990410158401015840)120583 119889119904
10158401015840
+ int1199041199041015840
1199041015840(119904 minus 11990410158401015840)120583 119889119904
10158401015840]
(46)
By denoting 119868(119904) = int1199040(1198891199041015840(119904 minus 1199041015840)120583) int1199041015840
0(11990410158401015840(119904 minus 11990410158401015840)120583)11988911990410158401015840 and
exchanging the order of quadratic integral 119868(119904) we obtain119868 (119904) = int119904
0
11988911990410158401015840(119904 minus 11990410158401015840)120583 int
119904
11990410158401015840
11990410158401015840(119904 minus 1199041015840)120583 119889119904
1015840 (47)
Thus
⟨1198832 (119904)⟩= 2
Γ2 (1 minus 120583) int119904
0
1199041015840(119904 minus 1199041015840)120583 119889119904
1015840 int1199041199041015840
1(119904 minus 11990410158401015840)120583 119889119904
10158401015840
= 2(1 minus 120583) Γ2 (1 minus 120583) int
119904
01199041015840 (119904 minus 1199041015840)1minus2120583 1198891199041015840
where 119870120583 = 2119861(2 2 minus 2120583)(1 minus 120583)Γ2(1 minus 120583) and 119861(119886 119887) =int10119909119886minus1(1 minus 119909)119887minus1119889119909 is Beta functionWe observe from (48) that in the limiting case 120583 rarr 0
memory function119872(119905) = 1 the parent process 119883(119904) definedby (43) reduces to the form defined by (11) and the secondmoment of 119883(119904) computed by (48) reduces to 11987001199043 where1198700 = 2119861(2 2) = 13 the same form as (16) We are alsointerested in the limiting case 120583 rarr 1 At the moment the
memory function 119872(119905) is a Dirac 120575-function 119883(119904) definedby (43) reduces to the standard Brownian motion
Let us turn to the MSD of the subordinated process 119909(119905)In terms of (18) and (48) we obtain
= 119870120583 Γ (4 minus 2120583)119906120572(3minus2120583)+1
(50)
Taking the inverse Laplace transform for ⟨(Δ119909)2⟩(119906) we have⟨(Δ119909)2⟩ (119905) = 119870120583120572119905120572(3minus2120583) 0 lt 120583 lt 1 0 lt 120572 le 1 (51)
where 119870120583120572 = 119870120583Γ(4 minus 2120583)Γ(120572(3 minus 2120583) + 1) In the limitingcase 120583 rarr 0 the parameter119870120583120572 reduces to
Thus (51) reduces to (25)It is easy to observe from (51) that there exists a compe-
tition between the memory parameter 120583 and stability index120572 For the case 120572 le 13 the subordinated process exhibitssubdiffusive behaviors For the case 13 lt 120572 lt 1 the processis subdiffusive when 1 lt 3minus2120583 lt 1120572 normal diffusive when3 minus 2120583 = 1120572 and superdiffusive when 1120572 lt 3 minus 2120583 lt 35 Conclusions
We introduce an integrated Brownian motion subordinatedby inverse 120572-stable one-sided Levy motion which is acontinuous limit of a jump-correlated CTRW In terms of theensembleMSD of the proposed process we conclude that theprocess is subdiffusive when 0 lt 120572 lt 13 normal diffusivewhen 120572 = 13 and superdiffusive when 13 lt 120572 le 1 Thetime-averagedMSD is also employed to showweak ergodicitybreaking occurring in the proposed process
We also generalize the process to the case where theparent process is fractional integral of Brownian motion Interms of the MSD we observe a competition between thememory parameter 120583 and stability index 120572 Other types ofinverse subordinators may be also candidates
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (Grant no 11671343) and the ScientificResearch Project of Hunan Provincial EducationDepartment(no 17B258)
6 Advances in Mathematical Physics
References
[1] J-P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990
[2] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[3] R Metzler and J Klafter ldquoThe restaurant at the end of therandomwalk recent developments in the description of anoma-lous transport by fractional dynamicsrdquo Journal of Physics AMathematical and General vol 37 no 31 pp R161ndashR208 2004
[4] I Eliazar and J Klafter ldquoAnomalous is ubiquitousrdquo Annals ofPhysics vol 326 no 9 pp 2517ndash2531 2011
[5] E W Montroll and G H Weiss ldquoRandom walks on lattices IIrdquoJournal of Mathematical Physics vol 6 pp 167ndash181 1965
[6] H C Fogedby ldquoLangevin equations for continuous time Levyflightsrdquo Physical Review E vol 50 no 2 pp 1657ndash1660 1994
[7] S Bochner ldquoDiffusion equation and stochastic processesrdquoProceedings of the National Academy of Sciences vol 35 no 7pp 368ndash370 1949
[8] A Baule and R Friedrich ldquoJoint probability distributions for aclass of non-Markovian processesrdquo Physical Review E vol 71no 2 Article ID 026101 2005
[9] A Piryatinska A I Saichev and W A Woyczynski ldquoModelsof anomalous diffusion the subdiffusive caserdquo Physica AStatistical Mechanics and Its Applications vol 349 no 3-4 pp375ndash420 2005
[10] M Magdziarz A Weron and K Weron ldquoFractional Fokker-Planck dynamics stochastic representation and computer sim-ulationrdquo Physical Review EmdashStatistical Nonlinear and SoftMatter Physics vol 75 no 1 Article ID 016708 2007
[11] D Kleinhans and R Friedrich ldquoContinuous-time randomwalks simulation of continuous trajectoriesrdquo Physical Review Evol 76 no 6 Article ID 061102 2007
[12] AWeronMMagdziarz and KWeron ldquoModeling of subdiffu-sion in space-time-dependent force fields beyond the fractionalFokker-Planck equationrdquo Physical Review E vol 77 no 3Article ID 036704 2008
[13] M Magdziarz A Weron and J Klafter ldquoEquivalence of thefractional fokker-planck and subordinated langevin equationsthe case of a time-dependent forcerdquo Physical Review Letters vol101 no 21 Article ID 210601 2008
[14] A Weron and S Orzel ldquoIto formula for subordinated Langevinequation A case of time dependent forcerdquoActa Physica PolonicaB vol 40 no 5 pp 1271ndash1277 2009
[15] S Eule and R Friedrich ldquoSubordinated Langevin equationsfor anomalous diffusion in external potentialsmdashbiasing anddecoupled external forcesrdquo EPL vol 86 no 3 Article ID 300082009
[16] M Magdziarz ldquoLangevin picture of subdiffusion with infinitelydivisiblewaiting timesrdquo Journal of Statistical Physics vol 135 no4 pp 763ndash772 2009
[17] M Magdziarz ldquoStochastic representation of subdiffusion pro-cesses with time-dependent driftrdquo Stochastic Processes andTheirApplications vol 119 no 10 pp 3238ndash3252 2009
[18] M Magdziarz and A Weron ldquoCompetition between subdiffu-sion and Levy flights a Monte Carlo approachrdquo Physical ReviewE vol 75 Article ID 056702 2007
[19] B o Dybiec and E Gudowska-Nowak ldquoSubordinated diffusionand continuous time random walk asymptoticsrdquo Chaos vol 20no 4 Article ID 043129 2010
[20] A Wyłomanska ldquoArithmetic Brownian motion subordinatedby tempered stable and inverse tempered stable processesrdquoPhysica A StatisticalMechanics and Its Applications vol 391 no22 pp 5685ndash5696 2012
[21] H Gu J-R Liang and Y-X Zhang ldquoOn a time-changedgeometric Brownian motion and its application in financialmarketrdquo Acta Physica Polonica B vol 43 no 8 pp 1667ndash16812012
[22] J Gajda and A Wyłomanska ldquoGeometric Brownian motionwith tempered stable waiting timesrdquo Journal of StatisticalPhysics vol 148 no 2 pp 296ndash305 2012
[23] J Janczura S Orzeł and A Wyłomanska ldquoSubordinated 120572120572-stable Ornstein-Uhlenbeck process as a tool for financial datadescriptionrdquo Physica A Statistical Mechanics and its Applica-tions vol 390 no 23-24 pp 4379ndash4387 2011
[24] J Gajda and AWyłomanska ldquoTime-changedOrnsteinndashUhlen-beck processrdquo Journal of Physics A Mathematical and Theoreti-cal vol 48 no 13 Article ID 135004 2015
[25] A Wyłomanska ldquoThe tempered stable process with infinitelydivisible inverse subordinatorsrdquo Journal of StatisticalMechanicsTheory and Experiment vol 2013 no 10 Article ID P10011 2013
[26] Y-X Zhang H Gu and J-R Liang ldquoFokker-planck typeequations associated with subordinated processes controlled bytempered 120572-stable processesrdquo Journal of Statistical Physics vol152 no 4 pp 742ndash752 2013
[27] J Gajda and A Wyłomanska ldquoFokkerndashPlanck type equationsassociated with fractional Brownian motion controlled byinfinitely divisible processesrdquo Physica A Statistical Mechanicsand Its Applications vol 405 pp 104ndash113 2014
[28] M Teuerle A Wyłomanska and G Sikora ldquoModeling anoma-lous diffusion by a subordinated fractional Levy-stable processrdquoJournal of Statistical Mechanics Theory and Experiment vol2013 no 5 Article ID P05016 2013
[29] J Gajda andMMagdziarz ldquoFractional Fokker-Planck equationwith tempered 120572-stable waiting times Langevin picture andcomputer simulationrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 011117 2010
[30] J Janczura and A Wyłomanska ldquoAnomalous diffusion mod-els different types of subordinator distributionrdquo Acta PhysicaPolonica B vol 43 no 5 pp 1001ndash1016 2012
[31] J Gajda ldquoFractional FokkerndashPlanck equationwith space depen-dent drift and diffusion the case of tempered 120572-stable waiting-timesrdquo Jagellonian University Institute of Physics Acta PhysicaPolonica B vol 44 no 5 pp 1149ndash1161 2013
[32] C Song T Koren P Wang and A Barabasi ldquoModelling thescaling properties of humanmobilityrdquoNature Physics vol 6 no10 pp 818ndash823 2010
[33] A Maye C-H Hsieh G Sugihara and B Brembs ldquoOrder inspontaneous behaviorrdquo PLoS ONE vol 2 no 5 article e4432007
[34] E Scalas ldquoThe application of continuous-time randomwalks infinance and economicsrdquo Physica A Statistical Mechanics and itsApplications vol 362 no 2 pp 225ndash239 2006
[35] A Chechkin M Hofmann and I M Sokolov ldquoContinuous-time random walk with correlated waiting timesrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 80no 3 Article ID 031112 2009
[36] M M Meerschaert E Nane and Y Xiao ldquoCorrelated continu-ous time random walksrdquo Statistics amp Probability Letters vol 79no 9 pp 1194ndash1202 2009
Advances in Mathematical Physics 7
[37] V Tejedor and R Metzler ldquoAnomalous diffusion in correlatedcontinuous time random walksrdquo Journal of Physics A Mathe-matical and Theoretical vol 43 no 8 Article ID 082002 2010
[38] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E vol 85 no 5 Article ID 051103 2012
[39] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time randomwalksmdashscaling limits andLangevin picturerdquo Journal of Statistical Mechanics Theory andExperiment vol 2012 no 4 Article ID P04010 2012
[40] M Magdziarz W Szczotka and P Zebrowski ldquoAsymptoticbehaviour of randomwalks with correlated temporal structurerdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 469 no 2159 Article ID 201304192013
[41] J H Schulz A V Chechkin and R Metzler ldquoCorrelatedcontinuous time random walks combining scale-invariancewith long-range memory for spatial and temporal dynamicsrdquoJournal of Physics A Mathematical and Theoretical vol 46 no47 Article ID 475001 2013
[42] L Shi Z Yu H Huang ZMao andA Xiao ldquoThe subordinatedprocesses controlled by a family of subordinators and corre-sponding FokkerndashPlanck type equationsrdquo Journal of StatisticalMechanics Theory and Experiment vol 2014 no 12 Article IDP12002 2014
[43] J Wang J Zhou L Lv W Qiu and F Ren ldquoHeterogeneousmemorized continuous time random walks in an external forcefieldsrdquo Journal of Statistical Physics vol 156 no 6 pp 1111ndash11242014
[44] F-Y Ren J Wang L-J Lv H Pan and W-Y Qiu ldquoEffect ofdifferent waiting time processes with memory to anomalousdiffusion dynamics in an external force fieldsrdquo Physica AStatistical Mechanics and Its Applications vol 417 pp 202ndash2142015
[45] L Lv F-Y Ren J Wang and J Xiao ldquoCorrelated continuoustime random walk with time averaged waiting timerdquo Physica AStatistical Mechanics and Its Applications vol 422 pp 101ndash1062015
[46] T W Burkhardt ldquoSemiflexible polymer in the half plane andstatistics of the integral of a Brownian curverdquo Journal of PhysicsA Mathematical and General vol 26 no 22 pp L1157ndashL11621993
[47] P Valageas ldquoStatistical properties of the burgers equation withbrownian initial velocityrdquo Journal of Statistical Physics vol 134no 3 pp 589ndash640 2009
[48] TW Burkhardt ldquoThe random acceleration process in boundedgeometriesrdquo Journal of Statistical Mechanics Theory and Exper-iment vol 2007 no 7 Article ID P07004 2007
[49] J Jeon A V Chechkin and R Metzler ldquoScaled Brownianmotion a paradoxical process with a time dependent diffusivityfor the description of anomalous diffusionrdquo Physical ChemistryChemical Physics vol 16 no 30 pp 15811ndash15817 2014
[50] Y He S Burov R Metzler and E Barkai ldquoRandom time-scaleinvariant diffusion and transport coefficientsrdquo Physical ReviewLetters vol 101 no 5 Article ID 058101 2008
[51] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 2009
[52] J-H Jeon and R Metzler ldquoFractional Brownian motion andmotion governed by the fractional Langevin equation in con-fined geometriesrdquo Physical Review E vol 81 no 2 Article ID021103 2010
[53] E Barkai Y Garini and R Metzler ldquoStrange kinetics of singlemolecules in living cellsrdquo Physics Today vol 65 no 8 pp 29ndash352012
[54] J Kursawe J Schulz and R Metzler ldquoTransient aging infractional Brownian and Langevin-equation motionrdquo PhysicalReview E vol 88 no 6 Article ID 062124 2013
[55] R Metzler J-H Jeon A G Cherstvy and E Barkai ldquoAnoma-lous diffusion models and their properties non-stationaritynon-ergodicity and ageing at the centenary of single particletrackingrdquo Physical Chemistry Chemical Physics vol 16 no 44pp 24128ndash24164 2014
[56] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
Assume that 119901(119909 119905) 119891(119909 119904) and 119892(119904 119905) are PDFs ofsubordinated process 119909(119905) parent process 119883(119904) and inversesubordinator 119904120572(119905) respectively In terms of subordinationwe have
It is easy to observe from (25) that the process issubdiffusive when 0 lt 120572 lt 13 normally diffusive when120572 = 13 and superdiffusive when 13 lt 120572 le 1
It is well-known that the MSD of the process given by (2)is of the form
Comparing (25) with (26) we see that Fogedbyrsquos model canonly represent anomalous subdiffusion but our model canrepresent subdiffusion normal diffusion and superdiffusion
Next we study weak ergodicity breaking of the subordi-nated process 119909(119905)
In an ergodic system one can find the equivalence
⟨(Δ119909)2⟩ (Δ) = ⟨1205752 (Δ)⟩ (27)
Here 1205752(Δ) is the time-averaged MSD of the process 119909(119905)defined as
1205752 (Δ) = 1119879 minus Δ int119879minusΔ
0[119909 (119905 + Δ) minus 119909 (119905)]2 119889119905 (28)
where Δ is the lag time and 119879 is the overall measure timeFor anomalous diffusion the behavior of the ensemble
MSD ⟨(Δ119909)2⟩(Δ) and the time-averaged MSD (28) may befundamentally different The disparity ⟨(Δ119909)2⟩(Δ) = ⟨1205752(Δ)⟩is usually calledweak ergodicity breaking (orweak nonergod-icity) [49] In recent years weak nonergodicity of anomalousdiffusion process attracts more and more attentions [49ndash55]
Since for any 119886 gt 0 parent process119883(119904) satisfies119883 (119886119904) = int119886119904
comparing (33) with (34) we see that the linear lag timedependence of ⟨1205752(Δ)⟩ is different from the power-law formΔ3120572 of ⟨(Δ119909)2⟩(Δ) which implies that subordinated process119909(119905) is weakly nonergodic
At last we consider the propagator 119901(119909 119905) associatedwith the subordinated process 119909(119905) By the total probabilityformula we obtain an integral representation of 119901(119909 119905)
where 0119868119901119905 is the Riemann-Liouville fractional integrationoperator of order 119901 defined by [56]
0119868119901119905 119891 (119905) = 1Γ (119901) int
119905
0(119905 minus 120591)119901minus1 119891 (120591) 119889120591 (119901 gt 0) (45)
As a result the jump-correlated CTRW has the subordi-nation form 119909(119905) = 119883(119904120572(119905)) where parent process 119883(119904) is ofthe form (44) and inverse subordinator 119904120572(119905) is defined by(12)
Here we are interested in the competition between thememory parameter 120583 and stability index 120572 In what followswe will not discuss any properties of motion other than theMSD
In terms of (44) we get
⟨1198832 (119904)⟩ = 1Γ2 (1 minus 120583)
sdot int1199040
1198891199041015840(119904 minus 1199041015840)120583 int
119904
0
⟨119861 (1199041015840) 119861 (11990410158401015840)⟩(119904 minus 11990410158401015840)120583 11988911990410158401015840
= 1Γ2 (1 minus 120583) int
119904
0
1198891199041015840(119904 minus 1199041015840)120583 int
119904
0
min 1199041015840 11990410158401015840(119904 minus 11990410158401015840)120583 11988911990410158401015840
= 1Γ2 (1 minus 120583) int
119904
0
1198891199041015840(119904 minus 1199041015840)120583 [int
1199041015840
0
11990410158401015840(119904 minus 11990410158401015840)120583 119889119904
10158401015840
+ int1199041199041015840
1199041015840(119904 minus 11990410158401015840)120583 119889119904
10158401015840]
(46)
By denoting 119868(119904) = int1199040(1198891199041015840(119904 minus 1199041015840)120583) int1199041015840
0(11990410158401015840(119904 minus 11990410158401015840)120583)11988911990410158401015840 and
exchanging the order of quadratic integral 119868(119904) we obtain119868 (119904) = int119904
0
11988911990410158401015840(119904 minus 11990410158401015840)120583 int
119904
11990410158401015840
11990410158401015840(119904 minus 1199041015840)120583 119889119904
1015840 (47)
Thus
⟨1198832 (119904)⟩= 2
Γ2 (1 minus 120583) int119904
0
1199041015840(119904 minus 1199041015840)120583 119889119904
1015840 int1199041199041015840
1(119904 minus 11990410158401015840)120583 119889119904
10158401015840
= 2(1 minus 120583) Γ2 (1 minus 120583) int
119904
01199041015840 (119904 minus 1199041015840)1minus2120583 1198891199041015840
where 119870120583 = 2119861(2 2 minus 2120583)(1 minus 120583)Γ2(1 minus 120583) and 119861(119886 119887) =int10119909119886minus1(1 minus 119909)119887minus1119889119909 is Beta functionWe observe from (48) that in the limiting case 120583 rarr 0
memory function119872(119905) = 1 the parent process 119883(119904) definedby (43) reduces to the form defined by (11) and the secondmoment of 119883(119904) computed by (48) reduces to 11987001199043 where1198700 = 2119861(2 2) = 13 the same form as (16) We are alsointerested in the limiting case 120583 rarr 1 At the moment the
memory function 119872(119905) is a Dirac 120575-function 119883(119904) definedby (43) reduces to the standard Brownian motion
Let us turn to the MSD of the subordinated process 119909(119905)In terms of (18) and (48) we obtain
= 119870120583 Γ (4 minus 2120583)119906120572(3minus2120583)+1
(50)
Taking the inverse Laplace transform for ⟨(Δ119909)2⟩(119906) we have⟨(Δ119909)2⟩ (119905) = 119870120583120572119905120572(3minus2120583) 0 lt 120583 lt 1 0 lt 120572 le 1 (51)
where 119870120583120572 = 119870120583Γ(4 minus 2120583)Γ(120572(3 minus 2120583) + 1) In the limitingcase 120583 rarr 0 the parameter119870120583120572 reduces to
Thus (51) reduces to (25)It is easy to observe from (51) that there exists a compe-
tition between the memory parameter 120583 and stability index120572 For the case 120572 le 13 the subordinated process exhibitssubdiffusive behaviors For the case 13 lt 120572 lt 1 the processis subdiffusive when 1 lt 3minus2120583 lt 1120572 normal diffusive when3 minus 2120583 = 1120572 and superdiffusive when 1120572 lt 3 minus 2120583 lt 35 Conclusions
We introduce an integrated Brownian motion subordinatedby inverse 120572-stable one-sided Levy motion which is acontinuous limit of a jump-correlated CTRW In terms of theensembleMSD of the proposed process we conclude that theprocess is subdiffusive when 0 lt 120572 lt 13 normal diffusivewhen 120572 = 13 and superdiffusive when 13 lt 120572 le 1 Thetime-averagedMSD is also employed to showweak ergodicitybreaking occurring in the proposed process
We also generalize the process to the case where theparent process is fractional integral of Brownian motion Interms of the MSD we observe a competition between thememory parameter 120583 and stability index 120572 Other types ofinverse subordinators may be also candidates
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (Grant no 11671343) and the ScientificResearch Project of Hunan Provincial EducationDepartment(no 17B258)
6 Advances in Mathematical Physics
References
[1] J-P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990
[2] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[3] R Metzler and J Klafter ldquoThe restaurant at the end of therandomwalk recent developments in the description of anoma-lous transport by fractional dynamicsrdquo Journal of Physics AMathematical and General vol 37 no 31 pp R161ndashR208 2004
[4] I Eliazar and J Klafter ldquoAnomalous is ubiquitousrdquo Annals ofPhysics vol 326 no 9 pp 2517ndash2531 2011
[5] E W Montroll and G H Weiss ldquoRandom walks on lattices IIrdquoJournal of Mathematical Physics vol 6 pp 167ndash181 1965
[6] H C Fogedby ldquoLangevin equations for continuous time Levyflightsrdquo Physical Review E vol 50 no 2 pp 1657ndash1660 1994
[7] S Bochner ldquoDiffusion equation and stochastic processesrdquoProceedings of the National Academy of Sciences vol 35 no 7pp 368ndash370 1949
[8] A Baule and R Friedrich ldquoJoint probability distributions for aclass of non-Markovian processesrdquo Physical Review E vol 71no 2 Article ID 026101 2005
[9] A Piryatinska A I Saichev and W A Woyczynski ldquoModelsof anomalous diffusion the subdiffusive caserdquo Physica AStatistical Mechanics and Its Applications vol 349 no 3-4 pp375ndash420 2005
[10] M Magdziarz A Weron and K Weron ldquoFractional Fokker-Planck dynamics stochastic representation and computer sim-ulationrdquo Physical Review EmdashStatistical Nonlinear and SoftMatter Physics vol 75 no 1 Article ID 016708 2007
[11] D Kleinhans and R Friedrich ldquoContinuous-time randomwalks simulation of continuous trajectoriesrdquo Physical Review Evol 76 no 6 Article ID 061102 2007
[12] AWeronMMagdziarz and KWeron ldquoModeling of subdiffu-sion in space-time-dependent force fields beyond the fractionalFokker-Planck equationrdquo Physical Review E vol 77 no 3Article ID 036704 2008
[13] M Magdziarz A Weron and J Klafter ldquoEquivalence of thefractional fokker-planck and subordinated langevin equationsthe case of a time-dependent forcerdquo Physical Review Letters vol101 no 21 Article ID 210601 2008
[14] A Weron and S Orzel ldquoIto formula for subordinated Langevinequation A case of time dependent forcerdquoActa Physica PolonicaB vol 40 no 5 pp 1271ndash1277 2009
[15] S Eule and R Friedrich ldquoSubordinated Langevin equationsfor anomalous diffusion in external potentialsmdashbiasing anddecoupled external forcesrdquo EPL vol 86 no 3 Article ID 300082009
[16] M Magdziarz ldquoLangevin picture of subdiffusion with infinitelydivisiblewaiting timesrdquo Journal of Statistical Physics vol 135 no4 pp 763ndash772 2009
[17] M Magdziarz ldquoStochastic representation of subdiffusion pro-cesses with time-dependent driftrdquo Stochastic Processes andTheirApplications vol 119 no 10 pp 3238ndash3252 2009
[18] M Magdziarz and A Weron ldquoCompetition between subdiffu-sion and Levy flights a Monte Carlo approachrdquo Physical ReviewE vol 75 Article ID 056702 2007
[19] B o Dybiec and E Gudowska-Nowak ldquoSubordinated diffusionand continuous time random walk asymptoticsrdquo Chaos vol 20no 4 Article ID 043129 2010
[20] A Wyłomanska ldquoArithmetic Brownian motion subordinatedby tempered stable and inverse tempered stable processesrdquoPhysica A StatisticalMechanics and Its Applications vol 391 no22 pp 5685ndash5696 2012
[21] H Gu J-R Liang and Y-X Zhang ldquoOn a time-changedgeometric Brownian motion and its application in financialmarketrdquo Acta Physica Polonica B vol 43 no 8 pp 1667ndash16812012
[22] J Gajda and A Wyłomanska ldquoGeometric Brownian motionwith tempered stable waiting timesrdquo Journal of StatisticalPhysics vol 148 no 2 pp 296ndash305 2012
[23] J Janczura S Orzeł and A Wyłomanska ldquoSubordinated 120572120572-stable Ornstein-Uhlenbeck process as a tool for financial datadescriptionrdquo Physica A Statistical Mechanics and its Applica-tions vol 390 no 23-24 pp 4379ndash4387 2011
[24] J Gajda and AWyłomanska ldquoTime-changedOrnsteinndashUhlen-beck processrdquo Journal of Physics A Mathematical and Theoreti-cal vol 48 no 13 Article ID 135004 2015
[25] A Wyłomanska ldquoThe tempered stable process with infinitelydivisible inverse subordinatorsrdquo Journal of StatisticalMechanicsTheory and Experiment vol 2013 no 10 Article ID P10011 2013
[26] Y-X Zhang H Gu and J-R Liang ldquoFokker-planck typeequations associated with subordinated processes controlled bytempered 120572-stable processesrdquo Journal of Statistical Physics vol152 no 4 pp 742ndash752 2013
[27] J Gajda and A Wyłomanska ldquoFokkerndashPlanck type equationsassociated with fractional Brownian motion controlled byinfinitely divisible processesrdquo Physica A Statistical Mechanicsand Its Applications vol 405 pp 104ndash113 2014
[28] M Teuerle A Wyłomanska and G Sikora ldquoModeling anoma-lous diffusion by a subordinated fractional Levy-stable processrdquoJournal of Statistical Mechanics Theory and Experiment vol2013 no 5 Article ID P05016 2013
[29] J Gajda andMMagdziarz ldquoFractional Fokker-Planck equationwith tempered 120572-stable waiting times Langevin picture andcomputer simulationrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 011117 2010
[30] J Janczura and A Wyłomanska ldquoAnomalous diffusion mod-els different types of subordinator distributionrdquo Acta PhysicaPolonica B vol 43 no 5 pp 1001ndash1016 2012
[31] J Gajda ldquoFractional FokkerndashPlanck equationwith space depen-dent drift and diffusion the case of tempered 120572-stable waiting-timesrdquo Jagellonian University Institute of Physics Acta PhysicaPolonica B vol 44 no 5 pp 1149ndash1161 2013
[32] C Song T Koren P Wang and A Barabasi ldquoModelling thescaling properties of humanmobilityrdquoNature Physics vol 6 no10 pp 818ndash823 2010
[33] A Maye C-H Hsieh G Sugihara and B Brembs ldquoOrder inspontaneous behaviorrdquo PLoS ONE vol 2 no 5 article e4432007
[34] E Scalas ldquoThe application of continuous-time randomwalks infinance and economicsrdquo Physica A Statistical Mechanics and itsApplications vol 362 no 2 pp 225ndash239 2006
[35] A Chechkin M Hofmann and I M Sokolov ldquoContinuous-time random walk with correlated waiting timesrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 80no 3 Article ID 031112 2009
[36] M M Meerschaert E Nane and Y Xiao ldquoCorrelated continu-ous time random walksrdquo Statistics amp Probability Letters vol 79no 9 pp 1194ndash1202 2009
Advances in Mathematical Physics 7
[37] V Tejedor and R Metzler ldquoAnomalous diffusion in correlatedcontinuous time random walksrdquo Journal of Physics A Mathe-matical and Theoretical vol 43 no 8 Article ID 082002 2010
[38] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E vol 85 no 5 Article ID 051103 2012
[39] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time randomwalksmdashscaling limits andLangevin picturerdquo Journal of Statistical Mechanics Theory andExperiment vol 2012 no 4 Article ID P04010 2012
[40] M Magdziarz W Szczotka and P Zebrowski ldquoAsymptoticbehaviour of randomwalks with correlated temporal structurerdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 469 no 2159 Article ID 201304192013
[41] J H Schulz A V Chechkin and R Metzler ldquoCorrelatedcontinuous time random walks combining scale-invariancewith long-range memory for spatial and temporal dynamicsrdquoJournal of Physics A Mathematical and Theoretical vol 46 no47 Article ID 475001 2013
[42] L Shi Z Yu H Huang ZMao andA Xiao ldquoThe subordinatedprocesses controlled by a family of subordinators and corre-sponding FokkerndashPlanck type equationsrdquo Journal of StatisticalMechanics Theory and Experiment vol 2014 no 12 Article IDP12002 2014
[43] J Wang J Zhou L Lv W Qiu and F Ren ldquoHeterogeneousmemorized continuous time random walks in an external forcefieldsrdquo Journal of Statistical Physics vol 156 no 6 pp 1111ndash11242014
[44] F-Y Ren J Wang L-J Lv H Pan and W-Y Qiu ldquoEffect ofdifferent waiting time processes with memory to anomalousdiffusion dynamics in an external force fieldsrdquo Physica AStatistical Mechanics and Its Applications vol 417 pp 202ndash2142015
[45] L Lv F-Y Ren J Wang and J Xiao ldquoCorrelated continuoustime random walk with time averaged waiting timerdquo Physica AStatistical Mechanics and Its Applications vol 422 pp 101ndash1062015
[46] T W Burkhardt ldquoSemiflexible polymer in the half plane andstatistics of the integral of a Brownian curverdquo Journal of PhysicsA Mathematical and General vol 26 no 22 pp L1157ndashL11621993
[47] P Valageas ldquoStatistical properties of the burgers equation withbrownian initial velocityrdquo Journal of Statistical Physics vol 134no 3 pp 589ndash640 2009
[48] TW Burkhardt ldquoThe random acceleration process in boundedgeometriesrdquo Journal of Statistical Mechanics Theory and Exper-iment vol 2007 no 7 Article ID P07004 2007
[49] J Jeon A V Chechkin and R Metzler ldquoScaled Brownianmotion a paradoxical process with a time dependent diffusivityfor the description of anomalous diffusionrdquo Physical ChemistryChemical Physics vol 16 no 30 pp 15811ndash15817 2014
[50] Y He S Burov R Metzler and E Barkai ldquoRandom time-scaleinvariant diffusion and transport coefficientsrdquo Physical ReviewLetters vol 101 no 5 Article ID 058101 2008
[51] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 2009
[52] J-H Jeon and R Metzler ldquoFractional Brownian motion andmotion governed by the fractional Langevin equation in con-fined geometriesrdquo Physical Review E vol 81 no 2 Article ID021103 2010
[53] E Barkai Y Garini and R Metzler ldquoStrange kinetics of singlemolecules in living cellsrdquo Physics Today vol 65 no 8 pp 29ndash352012
[54] J Kursawe J Schulz and R Metzler ldquoTransient aging infractional Brownian and Langevin-equation motionrdquo PhysicalReview E vol 88 no 6 Article ID 062124 2013
[55] R Metzler J-H Jeon A G Cherstvy and E Barkai ldquoAnoma-lous diffusion models and their properties non-stationaritynon-ergodicity and ageing at the centenary of single particletrackingrdquo Physical Chemistry Chemical Physics vol 16 no 44pp 24128ndash24164 2014
[56] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
comparing (33) with (34) we see that the linear lag timedependence of ⟨1205752(Δ)⟩ is different from the power-law formΔ3120572 of ⟨(Δ119909)2⟩(Δ) which implies that subordinated process119909(119905) is weakly nonergodic
At last we consider the propagator 119901(119909 119905) associatedwith the subordinated process 119909(119905) By the total probabilityformula we obtain an integral representation of 119901(119909 119905)
where 0119868119901119905 is the Riemann-Liouville fractional integrationoperator of order 119901 defined by [56]
0119868119901119905 119891 (119905) = 1Γ (119901) int
119905
0(119905 minus 120591)119901minus1 119891 (120591) 119889120591 (119901 gt 0) (45)
As a result the jump-correlated CTRW has the subordi-nation form 119909(119905) = 119883(119904120572(119905)) where parent process 119883(119904) is ofthe form (44) and inverse subordinator 119904120572(119905) is defined by(12)
Here we are interested in the competition between thememory parameter 120583 and stability index 120572 In what followswe will not discuss any properties of motion other than theMSD
In terms of (44) we get
⟨1198832 (119904)⟩ = 1Γ2 (1 minus 120583)
sdot int1199040
1198891199041015840(119904 minus 1199041015840)120583 int
119904
0
⟨119861 (1199041015840) 119861 (11990410158401015840)⟩(119904 minus 11990410158401015840)120583 11988911990410158401015840
= 1Γ2 (1 minus 120583) int
119904
0
1198891199041015840(119904 minus 1199041015840)120583 int
119904
0
min 1199041015840 11990410158401015840(119904 minus 11990410158401015840)120583 11988911990410158401015840
= 1Γ2 (1 minus 120583) int
119904
0
1198891199041015840(119904 minus 1199041015840)120583 [int
1199041015840
0
11990410158401015840(119904 minus 11990410158401015840)120583 119889119904
10158401015840
+ int1199041199041015840
1199041015840(119904 minus 11990410158401015840)120583 119889119904
10158401015840]
(46)
By denoting 119868(119904) = int1199040(1198891199041015840(119904 minus 1199041015840)120583) int1199041015840
0(11990410158401015840(119904 minus 11990410158401015840)120583)11988911990410158401015840 and
exchanging the order of quadratic integral 119868(119904) we obtain119868 (119904) = int119904
0
11988911990410158401015840(119904 minus 11990410158401015840)120583 int
119904
11990410158401015840
11990410158401015840(119904 minus 1199041015840)120583 119889119904
1015840 (47)
Thus
⟨1198832 (119904)⟩= 2
Γ2 (1 minus 120583) int119904
0
1199041015840(119904 minus 1199041015840)120583 119889119904
1015840 int1199041199041015840
1(119904 minus 11990410158401015840)120583 119889119904
10158401015840
= 2(1 minus 120583) Γ2 (1 minus 120583) int
119904
01199041015840 (119904 minus 1199041015840)1minus2120583 1198891199041015840
where 119870120583 = 2119861(2 2 minus 2120583)(1 minus 120583)Γ2(1 minus 120583) and 119861(119886 119887) =int10119909119886minus1(1 minus 119909)119887minus1119889119909 is Beta functionWe observe from (48) that in the limiting case 120583 rarr 0
memory function119872(119905) = 1 the parent process 119883(119904) definedby (43) reduces to the form defined by (11) and the secondmoment of 119883(119904) computed by (48) reduces to 11987001199043 where1198700 = 2119861(2 2) = 13 the same form as (16) We are alsointerested in the limiting case 120583 rarr 1 At the moment the
memory function 119872(119905) is a Dirac 120575-function 119883(119904) definedby (43) reduces to the standard Brownian motion
Let us turn to the MSD of the subordinated process 119909(119905)In terms of (18) and (48) we obtain
= 119870120583 Γ (4 minus 2120583)119906120572(3minus2120583)+1
(50)
Taking the inverse Laplace transform for ⟨(Δ119909)2⟩(119906) we have⟨(Δ119909)2⟩ (119905) = 119870120583120572119905120572(3minus2120583) 0 lt 120583 lt 1 0 lt 120572 le 1 (51)
where 119870120583120572 = 119870120583Γ(4 minus 2120583)Γ(120572(3 minus 2120583) + 1) In the limitingcase 120583 rarr 0 the parameter119870120583120572 reduces to
Thus (51) reduces to (25)It is easy to observe from (51) that there exists a compe-
tition between the memory parameter 120583 and stability index120572 For the case 120572 le 13 the subordinated process exhibitssubdiffusive behaviors For the case 13 lt 120572 lt 1 the processis subdiffusive when 1 lt 3minus2120583 lt 1120572 normal diffusive when3 minus 2120583 = 1120572 and superdiffusive when 1120572 lt 3 minus 2120583 lt 35 Conclusions
We introduce an integrated Brownian motion subordinatedby inverse 120572-stable one-sided Levy motion which is acontinuous limit of a jump-correlated CTRW In terms of theensembleMSD of the proposed process we conclude that theprocess is subdiffusive when 0 lt 120572 lt 13 normal diffusivewhen 120572 = 13 and superdiffusive when 13 lt 120572 le 1 Thetime-averagedMSD is also employed to showweak ergodicitybreaking occurring in the proposed process
We also generalize the process to the case where theparent process is fractional integral of Brownian motion Interms of the MSD we observe a competition between thememory parameter 120583 and stability index 120572 Other types ofinverse subordinators may be also candidates
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (Grant no 11671343) and the ScientificResearch Project of Hunan Provincial EducationDepartment(no 17B258)
6 Advances in Mathematical Physics
References
[1] J-P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990
[2] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[3] R Metzler and J Klafter ldquoThe restaurant at the end of therandomwalk recent developments in the description of anoma-lous transport by fractional dynamicsrdquo Journal of Physics AMathematical and General vol 37 no 31 pp R161ndashR208 2004
[4] I Eliazar and J Klafter ldquoAnomalous is ubiquitousrdquo Annals ofPhysics vol 326 no 9 pp 2517ndash2531 2011
[5] E W Montroll and G H Weiss ldquoRandom walks on lattices IIrdquoJournal of Mathematical Physics vol 6 pp 167ndash181 1965
[6] H C Fogedby ldquoLangevin equations for continuous time Levyflightsrdquo Physical Review E vol 50 no 2 pp 1657ndash1660 1994
[7] S Bochner ldquoDiffusion equation and stochastic processesrdquoProceedings of the National Academy of Sciences vol 35 no 7pp 368ndash370 1949
[8] A Baule and R Friedrich ldquoJoint probability distributions for aclass of non-Markovian processesrdquo Physical Review E vol 71no 2 Article ID 026101 2005
[9] A Piryatinska A I Saichev and W A Woyczynski ldquoModelsof anomalous diffusion the subdiffusive caserdquo Physica AStatistical Mechanics and Its Applications vol 349 no 3-4 pp375ndash420 2005
[10] M Magdziarz A Weron and K Weron ldquoFractional Fokker-Planck dynamics stochastic representation and computer sim-ulationrdquo Physical Review EmdashStatistical Nonlinear and SoftMatter Physics vol 75 no 1 Article ID 016708 2007
[11] D Kleinhans and R Friedrich ldquoContinuous-time randomwalks simulation of continuous trajectoriesrdquo Physical Review Evol 76 no 6 Article ID 061102 2007
[12] AWeronMMagdziarz and KWeron ldquoModeling of subdiffu-sion in space-time-dependent force fields beyond the fractionalFokker-Planck equationrdquo Physical Review E vol 77 no 3Article ID 036704 2008
[13] M Magdziarz A Weron and J Klafter ldquoEquivalence of thefractional fokker-planck and subordinated langevin equationsthe case of a time-dependent forcerdquo Physical Review Letters vol101 no 21 Article ID 210601 2008
[14] A Weron and S Orzel ldquoIto formula for subordinated Langevinequation A case of time dependent forcerdquoActa Physica PolonicaB vol 40 no 5 pp 1271ndash1277 2009
[15] S Eule and R Friedrich ldquoSubordinated Langevin equationsfor anomalous diffusion in external potentialsmdashbiasing anddecoupled external forcesrdquo EPL vol 86 no 3 Article ID 300082009
[16] M Magdziarz ldquoLangevin picture of subdiffusion with infinitelydivisiblewaiting timesrdquo Journal of Statistical Physics vol 135 no4 pp 763ndash772 2009
[17] M Magdziarz ldquoStochastic representation of subdiffusion pro-cesses with time-dependent driftrdquo Stochastic Processes andTheirApplications vol 119 no 10 pp 3238ndash3252 2009
[18] M Magdziarz and A Weron ldquoCompetition between subdiffu-sion and Levy flights a Monte Carlo approachrdquo Physical ReviewE vol 75 Article ID 056702 2007
[19] B o Dybiec and E Gudowska-Nowak ldquoSubordinated diffusionand continuous time random walk asymptoticsrdquo Chaos vol 20no 4 Article ID 043129 2010
[20] A Wyłomanska ldquoArithmetic Brownian motion subordinatedby tempered stable and inverse tempered stable processesrdquoPhysica A StatisticalMechanics and Its Applications vol 391 no22 pp 5685ndash5696 2012
[21] H Gu J-R Liang and Y-X Zhang ldquoOn a time-changedgeometric Brownian motion and its application in financialmarketrdquo Acta Physica Polonica B vol 43 no 8 pp 1667ndash16812012
[22] J Gajda and A Wyłomanska ldquoGeometric Brownian motionwith tempered stable waiting timesrdquo Journal of StatisticalPhysics vol 148 no 2 pp 296ndash305 2012
[23] J Janczura S Orzeł and A Wyłomanska ldquoSubordinated 120572120572-stable Ornstein-Uhlenbeck process as a tool for financial datadescriptionrdquo Physica A Statistical Mechanics and its Applica-tions vol 390 no 23-24 pp 4379ndash4387 2011
[24] J Gajda and AWyłomanska ldquoTime-changedOrnsteinndashUhlen-beck processrdquo Journal of Physics A Mathematical and Theoreti-cal vol 48 no 13 Article ID 135004 2015
[25] A Wyłomanska ldquoThe tempered stable process with infinitelydivisible inverse subordinatorsrdquo Journal of StatisticalMechanicsTheory and Experiment vol 2013 no 10 Article ID P10011 2013
[26] Y-X Zhang H Gu and J-R Liang ldquoFokker-planck typeequations associated with subordinated processes controlled bytempered 120572-stable processesrdquo Journal of Statistical Physics vol152 no 4 pp 742ndash752 2013
[27] J Gajda and A Wyłomanska ldquoFokkerndashPlanck type equationsassociated with fractional Brownian motion controlled byinfinitely divisible processesrdquo Physica A Statistical Mechanicsand Its Applications vol 405 pp 104ndash113 2014
[28] M Teuerle A Wyłomanska and G Sikora ldquoModeling anoma-lous diffusion by a subordinated fractional Levy-stable processrdquoJournal of Statistical Mechanics Theory and Experiment vol2013 no 5 Article ID P05016 2013
[29] J Gajda andMMagdziarz ldquoFractional Fokker-Planck equationwith tempered 120572-stable waiting times Langevin picture andcomputer simulationrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 011117 2010
[30] J Janczura and A Wyłomanska ldquoAnomalous diffusion mod-els different types of subordinator distributionrdquo Acta PhysicaPolonica B vol 43 no 5 pp 1001ndash1016 2012
[31] J Gajda ldquoFractional FokkerndashPlanck equationwith space depen-dent drift and diffusion the case of tempered 120572-stable waiting-timesrdquo Jagellonian University Institute of Physics Acta PhysicaPolonica B vol 44 no 5 pp 1149ndash1161 2013
[32] C Song T Koren P Wang and A Barabasi ldquoModelling thescaling properties of humanmobilityrdquoNature Physics vol 6 no10 pp 818ndash823 2010
[33] A Maye C-H Hsieh G Sugihara and B Brembs ldquoOrder inspontaneous behaviorrdquo PLoS ONE vol 2 no 5 article e4432007
[34] E Scalas ldquoThe application of continuous-time randomwalks infinance and economicsrdquo Physica A Statistical Mechanics and itsApplications vol 362 no 2 pp 225ndash239 2006
[35] A Chechkin M Hofmann and I M Sokolov ldquoContinuous-time random walk with correlated waiting timesrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 80no 3 Article ID 031112 2009
[36] M M Meerschaert E Nane and Y Xiao ldquoCorrelated continu-ous time random walksrdquo Statistics amp Probability Letters vol 79no 9 pp 1194ndash1202 2009
Advances in Mathematical Physics 7
[37] V Tejedor and R Metzler ldquoAnomalous diffusion in correlatedcontinuous time random walksrdquo Journal of Physics A Mathe-matical and Theoretical vol 43 no 8 Article ID 082002 2010
[38] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E vol 85 no 5 Article ID 051103 2012
[39] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time randomwalksmdashscaling limits andLangevin picturerdquo Journal of Statistical Mechanics Theory andExperiment vol 2012 no 4 Article ID P04010 2012
[40] M Magdziarz W Szczotka and P Zebrowski ldquoAsymptoticbehaviour of randomwalks with correlated temporal structurerdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 469 no 2159 Article ID 201304192013
[41] J H Schulz A V Chechkin and R Metzler ldquoCorrelatedcontinuous time random walks combining scale-invariancewith long-range memory for spatial and temporal dynamicsrdquoJournal of Physics A Mathematical and Theoretical vol 46 no47 Article ID 475001 2013
[42] L Shi Z Yu H Huang ZMao andA Xiao ldquoThe subordinatedprocesses controlled by a family of subordinators and corre-sponding FokkerndashPlanck type equationsrdquo Journal of StatisticalMechanics Theory and Experiment vol 2014 no 12 Article IDP12002 2014
[43] J Wang J Zhou L Lv W Qiu and F Ren ldquoHeterogeneousmemorized continuous time random walks in an external forcefieldsrdquo Journal of Statistical Physics vol 156 no 6 pp 1111ndash11242014
[44] F-Y Ren J Wang L-J Lv H Pan and W-Y Qiu ldquoEffect ofdifferent waiting time processes with memory to anomalousdiffusion dynamics in an external force fieldsrdquo Physica AStatistical Mechanics and Its Applications vol 417 pp 202ndash2142015
[45] L Lv F-Y Ren J Wang and J Xiao ldquoCorrelated continuoustime random walk with time averaged waiting timerdquo Physica AStatistical Mechanics and Its Applications vol 422 pp 101ndash1062015
[46] T W Burkhardt ldquoSemiflexible polymer in the half plane andstatistics of the integral of a Brownian curverdquo Journal of PhysicsA Mathematical and General vol 26 no 22 pp L1157ndashL11621993
[47] P Valageas ldquoStatistical properties of the burgers equation withbrownian initial velocityrdquo Journal of Statistical Physics vol 134no 3 pp 589ndash640 2009
[48] TW Burkhardt ldquoThe random acceleration process in boundedgeometriesrdquo Journal of Statistical Mechanics Theory and Exper-iment vol 2007 no 7 Article ID P07004 2007
[49] J Jeon A V Chechkin and R Metzler ldquoScaled Brownianmotion a paradoxical process with a time dependent diffusivityfor the description of anomalous diffusionrdquo Physical ChemistryChemical Physics vol 16 no 30 pp 15811ndash15817 2014
[50] Y He S Burov R Metzler and E Barkai ldquoRandom time-scaleinvariant diffusion and transport coefficientsrdquo Physical ReviewLetters vol 101 no 5 Article ID 058101 2008
[51] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 2009
[52] J-H Jeon and R Metzler ldquoFractional Brownian motion andmotion governed by the fractional Langevin equation in con-fined geometriesrdquo Physical Review E vol 81 no 2 Article ID021103 2010
[53] E Barkai Y Garini and R Metzler ldquoStrange kinetics of singlemolecules in living cellsrdquo Physics Today vol 65 no 8 pp 29ndash352012
[54] J Kursawe J Schulz and R Metzler ldquoTransient aging infractional Brownian and Langevin-equation motionrdquo PhysicalReview E vol 88 no 6 Article ID 062124 2013
[55] R Metzler J-H Jeon A G Cherstvy and E Barkai ldquoAnoma-lous diffusion models and their properties non-stationaritynon-ergodicity and ageing at the centenary of single particletrackingrdquo Physical Chemistry Chemical Physics vol 16 no 44pp 24128ndash24164 2014
[56] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
where 0119868119901119905 is the Riemann-Liouville fractional integrationoperator of order 119901 defined by [56]
0119868119901119905 119891 (119905) = 1Γ (119901) int
119905
0(119905 minus 120591)119901minus1 119891 (120591) 119889120591 (119901 gt 0) (45)
As a result the jump-correlated CTRW has the subordi-nation form 119909(119905) = 119883(119904120572(119905)) where parent process 119883(119904) is ofthe form (44) and inverse subordinator 119904120572(119905) is defined by(12)
Here we are interested in the competition between thememory parameter 120583 and stability index 120572 In what followswe will not discuss any properties of motion other than theMSD
In terms of (44) we get
⟨1198832 (119904)⟩ = 1Γ2 (1 minus 120583)
sdot int1199040
1198891199041015840(119904 minus 1199041015840)120583 int
119904
0
⟨119861 (1199041015840) 119861 (11990410158401015840)⟩(119904 minus 11990410158401015840)120583 11988911990410158401015840
= 1Γ2 (1 minus 120583) int
119904
0
1198891199041015840(119904 minus 1199041015840)120583 int
119904
0
min 1199041015840 11990410158401015840(119904 minus 11990410158401015840)120583 11988911990410158401015840
= 1Γ2 (1 minus 120583) int
119904
0
1198891199041015840(119904 minus 1199041015840)120583 [int
1199041015840
0
11990410158401015840(119904 minus 11990410158401015840)120583 119889119904
10158401015840
+ int1199041199041015840
1199041015840(119904 minus 11990410158401015840)120583 119889119904
10158401015840]
(46)
By denoting 119868(119904) = int1199040(1198891199041015840(119904 minus 1199041015840)120583) int1199041015840
0(11990410158401015840(119904 minus 11990410158401015840)120583)11988911990410158401015840 and
exchanging the order of quadratic integral 119868(119904) we obtain119868 (119904) = int119904
0
11988911990410158401015840(119904 minus 11990410158401015840)120583 int
119904
11990410158401015840
11990410158401015840(119904 minus 1199041015840)120583 119889119904
1015840 (47)
Thus
⟨1198832 (119904)⟩= 2
Γ2 (1 minus 120583) int119904
0
1199041015840(119904 minus 1199041015840)120583 119889119904
1015840 int1199041199041015840
1(119904 minus 11990410158401015840)120583 119889119904
10158401015840
= 2(1 minus 120583) Γ2 (1 minus 120583) int
119904
01199041015840 (119904 minus 1199041015840)1minus2120583 1198891199041015840
where 119870120583 = 2119861(2 2 minus 2120583)(1 minus 120583)Γ2(1 minus 120583) and 119861(119886 119887) =int10119909119886minus1(1 minus 119909)119887minus1119889119909 is Beta functionWe observe from (48) that in the limiting case 120583 rarr 0
memory function119872(119905) = 1 the parent process 119883(119904) definedby (43) reduces to the form defined by (11) and the secondmoment of 119883(119904) computed by (48) reduces to 11987001199043 where1198700 = 2119861(2 2) = 13 the same form as (16) We are alsointerested in the limiting case 120583 rarr 1 At the moment the
memory function 119872(119905) is a Dirac 120575-function 119883(119904) definedby (43) reduces to the standard Brownian motion
Let us turn to the MSD of the subordinated process 119909(119905)In terms of (18) and (48) we obtain
= 119870120583 Γ (4 minus 2120583)119906120572(3minus2120583)+1
(50)
Taking the inverse Laplace transform for ⟨(Δ119909)2⟩(119906) we have⟨(Δ119909)2⟩ (119905) = 119870120583120572119905120572(3minus2120583) 0 lt 120583 lt 1 0 lt 120572 le 1 (51)
where 119870120583120572 = 119870120583Γ(4 minus 2120583)Γ(120572(3 minus 2120583) + 1) In the limitingcase 120583 rarr 0 the parameter119870120583120572 reduces to
Thus (51) reduces to (25)It is easy to observe from (51) that there exists a compe-
tition between the memory parameter 120583 and stability index120572 For the case 120572 le 13 the subordinated process exhibitssubdiffusive behaviors For the case 13 lt 120572 lt 1 the processis subdiffusive when 1 lt 3minus2120583 lt 1120572 normal diffusive when3 minus 2120583 = 1120572 and superdiffusive when 1120572 lt 3 minus 2120583 lt 35 Conclusions
We introduce an integrated Brownian motion subordinatedby inverse 120572-stable one-sided Levy motion which is acontinuous limit of a jump-correlated CTRW In terms of theensembleMSD of the proposed process we conclude that theprocess is subdiffusive when 0 lt 120572 lt 13 normal diffusivewhen 120572 = 13 and superdiffusive when 13 lt 120572 le 1 Thetime-averagedMSD is also employed to showweak ergodicitybreaking occurring in the proposed process
We also generalize the process to the case where theparent process is fractional integral of Brownian motion Interms of the MSD we observe a competition between thememory parameter 120583 and stability index 120572 Other types ofinverse subordinators may be also candidates
Conflicts of Interest
The authors declare that there are no conflicts of interestregarding the publication of this paper
Acknowledgments
This work is supported by the National Natural ScienceFoundation of China (Grant no 11671343) and the ScientificResearch Project of Hunan Provincial EducationDepartment(no 17B258)
6 Advances in Mathematical Physics
References
[1] J-P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990
[2] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[3] R Metzler and J Klafter ldquoThe restaurant at the end of therandomwalk recent developments in the description of anoma-lous transport by fractional dynamicsrdquo Journal of Physics AMathematical and General vol 37 no 31 pp R161ndashR208 2004
[4] I Eliazar and J Klafter ldquoAnomalous is ubiquitousrdquo Annals ofPhysics vol 326 no 9 pp 2517ndash2531 2011
[5] E W Montroll and G H Weiss ldquoRandom walks on lattices IIrdquoJournal of Mathematical Physics vol 6 pp 167ndash181 1965
[6] H C Fogedby ldquoLangevin equations for continuous time Levyflightsrdquo Physical Review E vol 50 no 2 pp 1657ndash1660 1994
[7] S Bochner ldquoDiffusion equation and stochastic processesrdquoProceedings of the National Academy of Sciences vol 35 no 7pp 368ndash370 1949
[8] A Baule and R Friedrich ldquoJoint probability distributions for aclass of non-Markovian processesrdquo Physical Review E vol 71no 2 Article ID 026101 2005
[9] A Piryatinska A I Saichev and W A Woyczynski ldquoModelsof anomalous diffusion the subdiffusive caserdquo Physica AStatistical Mechanics and Its Applications vol 349 no 3-4 pp375ndash420 2005
[10] M Magdziarz A Weron and K Weron ldquoFractional Fokker-Planck dynamics stochastic representation and computer sim-ulationrdquo Physical Review EmdashStatistical Nonlinear and SoftMatter Physics vol 75 no 1 Article ID 016708 2007
[11] D Kleinhans and R Friedrich ldquoContinuous-time randomwalks simulation of continuous trajectoriesrdquo Physical Review Evol 76 no 6 Article ID 061102 2007
[12] AWeronMMagdziarz and KWeron ldquoModeling of subdiffu-sion in space-time-dependent force fields beyond the fractionalFokker-Planck equationrdquo Physical Review E vol 77 no 3Article ID 036704 2008
[13] M Magdziarz A Weron and J Klafter ldquoEquivalence of thefractional fokker-planck and subordinated langevin equationsthe case of a time-dependent forcerdquo Physical Review Letters vol101 no 21 Article ID 210601 2008
[14] A Weron and S Orzel ldquoIto formula for subordinated Langevinequation A case of time dependent forcerdquoActa Physica PolonicaB vol 40 no 5 pp 1271ndash1277 2009
[15] S Eule and R Friedrich ldquoSubordinated Langevin equationsfor anomalous diffusion in external potentialsmdashbiasing anddecoupled external forcesrdquo EPL vol 86 no 3 Article ID 300082009
[16] M Magdziarz ldquoLangevin picture of subdiffusion with infinitelydivisiblewaiting timesrdquo Journal of Statistical Physics vol 135 no4 pp 763ndash772 2009
[17] M Magdziarz ldquoStochastic representation of subdiffusion pro-cesses with time-dependent driftrdquo Stochastic Processes andTheirApplications vol 119 no 10 pp 3238ndash3252 2009
[18] M Magdziarz and A Weron ldquoCompetition between subdiffu-sion and Levy flights a Monte Carlo approachrdquo Physical ReviewE vol 75 Article ID 056702 2007
[19] B o Dybiec and E Gudowska-Nowak ldquoSubordinated diffusionand continuous time random walk asymptoticsrdquo Chaos vol 20no 4 Article ID 043129 2010
[20] A Wyłomanska ldquoArithmetic Brownian motion subordinatedby tempered stable and inverse tempered stable processesrdquoPhysica A StatisticalMechanics and Its Applications vol 391 no22 pp 5685ndash5696 2012
[21] H Gu J-R Liang and Y-X Zhang ldquoOn a time-changedgeometric Brownian motion and its application in financialmarketrdquo Acta Physica Polonica B vol 43 no 8 pp 1667ndash16812012
[22] J Gajda and A Wyłomanska ldquoGeometric Brownian motionwith tempered stable waiting timesrdquo Journal of StatisticalPhysics vol 148 no 2 pp 296ndash305 2012
[23] J Janczura S Orzeł and A Wyłomanska ldquoSubordinated 120572120572-stable Ornstein-Uhlenbeck process as a tool for financial datadescriptionrdquo Physica A Statistical Mechanics and its Applica-tions vol 390 no 23-24 pp 4379ndash4387 2011
[24] J Gajda and AWyłomanska ldquoTime-changedOrnsteinndashUhlen-beck processrdquo Journal of Physics A Mathematical and Theoreti-cal vol 48 no 13 Article ID 135004 2015
[25] A Wyłomanska ldquoThe tempered stable process with infinitelydivisible inverse subordinatorsrdquo Journal of StatisticalMechanicsTheory and Experiment vol 2013 no 10 Article ID P10011 2013
[26] Y-X Zhang H Gu and J-R Liang ldquoFokker-planck typeequations associated with subordinated processes controlled bytempered 120572-stable processesrdquo Journal of Statistical Physics vol152 no 4 pp 742ndash752 2013
[27] J Gajda and A Wyłomanska ldquoFokkerndashPlanck type equationsassociated with fractional Brownian motion controlled byinfinitely divisible processesrdquo Physica A Statistical Mechanicsand Its Applications vol 405 pp 104ndash113 2014
[28] M Teuerle A Wyłomanska and G Sikora ldquoModeling anoma-lous diffusion by a subordinated fractional Levy-stable processrdquoJournal of Statistical Mechanics Theory and Experiment vol2013 no 5 Article ID P05016 2013
[29] J Gajda andMMagdziarz ldquoFractional Fokker-Planck equationwith tempered 120572-stable waiting times Langevin picture andcomputer simulationrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 011117 2010
[30] J Janczura and A Wyłomanska ldquoAnomalous diffusion mod-els different types of subordinator distributionrdquo Acta PhysicaPolonica B vol 43 no 5 pp 1001ndash1016 2012
[31] J Gajda ldquoFractional FokkerndashPlanck equationwith space depen-dent drift and diffusion the case of tempered 120572-stable waiting-timesrdquo Jagellonian University Institute of Physics Acta PhysicaPolonica B vol 44 no 5 pp 1149ndash1161 2013
[32] C Song T Koren P Wang and A Barabasi ldquoModelling thescaling properties of humanmobilityrdquoNature Physics vol 6 no10 pp 818ndash823 2010
[33] A Maye C-H Hsieh G Sugihara and B Brembs ldquoOrder inspontaneous behaviorrdquo PLoS ONE vol 2 no 5 article e4432007
[34] E Scalas ldquoThe application of continuous-time randomwalks infinance and economicsrdquo Physica A Statistical Mechanics and itsApplications vol 362 no 2 pp 225ndash239 2006
[35] A Chechkin M Hofmann and I M Sokolov ldquoContinuous-time random walk with correlated waiting timesrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 80no 3 Article ID 031112 2009
[36] M M Meerschaert E Nane and Y Xiao ldquoCorrelated continu-ous time random walksrdquo Statistics amp Probability Letters vol 79no 9 pp 1194ndash1202 2009
Advances in Mathematical Physics 7
[37] V Tejedor and R Metzler ldquoAnomalous diffusion in correlatedcontinuous time random walksrdquo Journal of Physics A Mathe-matical and Theoretical vol 43 no 8 Article ID 082002 2010
[38] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E vol 85 no 5 Article ID 051103 2012
[39] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time randomwalksmdashscaling limits andLangevin picturerdquo Journal of Statistical Mechanics Theory andExperiment vol 2012 no 4 Article ID P04010 2012
[40] M Magdziarz W Szczotka and P Zebrowski ldquoAsymptoticbehaviour of randomwalks with correlated temporal structurerdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 469 no 2159 Article ID 201304192013
[41] J H Schulz A V Chechkin and R Metzler ldquoCorrelatedcontinuous time random walks combining scale-invariancewith long-range memory for spatial and temporal dynamicsrdquoJournal of Physics A Mathematical and Theoretical vol 46 no47 Article ID 475001 2013
[42] L Shi Z Yu H Huang ZMao andA Xiao ldquoThe subordinatedprocesses controlled by a family of subordinators and corre-sponding FokkerndashPlanck type equationsrdquo Journal of StatisticalMechanics Theory and Experiment vol 2014 no 12 Article IDP12002 2014
[43] J Wang J Zhou L Lv W Qiu and F Ren ldquoHeterogeneousmemorized continuous time random walks in an external forcefieldsrdquo Journal of Statistical Physics vol 156 no 6 pp 1111ndash11242014
[44] F-Y Ren J Wang L-J Lv H Pan and W-Y Qiu ldquoEffect ofdifferent waiting time processes with memory to anomalousdiffusion dynamics in an external force fieldsrdquo Physica AStatistical Mechanics and Its Applications vol 417 pp 202ndash2142015
[45] L Lv F-Y Ren J Wang and J Xiao ldquoCorrelated continuoustime random walk with time averaged waiting timerdquo Physica AStatistical Mechanics and Its Applications vol 422 pp 101ndash1062015
[46] T W Burkhardt ldquoSemiflexible polymer in the half plane andstatistics of the integral of a Brownian curverdquo Journal of PhysicsA Mathematical and General vol 26 no 22 pp L1157ndashL11621993
[47] P Valageas ldquoStatistical properties of the burgers equation withbrownian initial velocityrdquo Journal of Statistical Physics vol 134no 3 pp 589ndash640 2009
[48] TW Burkhardt ldquoThe random acceleration process in boundedgeometriesrdquo Journal of Statistical Mechanics Theory and Exper-iment vol 2007 no 7 Article ID P07004 2007
[49] J Jeon A V Chechkin and R Metzler ldquoScaled Brownianmotion a paradoxical process with a time dependent diffusivityfor the description of anomalous diffusionrdquo Physical ChemistryChemical Physics vol 16 no 30 pp 15811ndash15817 2014
[50] Y He S Burov R Metzler and E Barkai ldquoRandom time-scaleinvariant diffusion and transport coefficientsrdquo Physical ReviewLetters vol 101 no 5 Article ID 058101 2008
[51] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 2009
[52] J-H Jeon and R Metzler ldquoFractional Brownian motion andmotion governed by the fractional Langevin equation in con-fined geometriesrdquo Physical Review E vol 81 no 2 Article ID021103 2010
[53] E Barkai Y Garini and R Metzler ldquoStrange kinetics of singlemolecules in living cellsrdquo Physics Today vol 65 no 8 pp 29ndash352012
[54] J Kursawe J Schulz and R Metzler ldquoTransient aging infractional Brownian and Langevin-equation motionrdquo PhysicalReview E vol 88 no 6 Article ID 062124 2013
[55] R Metzler J-H Jeon A G Cherstvy and E Barkai ldquoAnoma-lous diffusion models and their properties non-stationaritynon-ergodicity and ageing at the centenary of single particletrackingrdquo Physical Chemistry Chemical Physics vol 16 no 44pp 24128ndash24164 2014
[56] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[1] J-P Bouchaud and A Georges ldquoAnomalous diffusion in dis-ordered media statistical mechanisms models and physicalapplicationsrdquoPhysics Reports vol 195 no 4-5 pp 127ndash293 1990
[2] R Metzler and J Klafter ldquoThe random walkrsquos guide to anoma-lous diffusion a fractional dynamics approachrdquo Physics Reportsvol 339 no 1 pp 1ndash77 2000
[3] R Metzler and J Klafter ldquoThe restaurant at the end of therandomwalk recent developments in the description of anoma-lous transport by fractional dynamicsrdquo Journal of Physics AMathematical and General vol 37 no 31 pp R161ndashR208 2004
[4] I Eliazar and J Klafter ldquoAnomalous is ubiquitousrdquo Annals ofPhysics vol 326 no 9 pp 2517ndash2531 2011
[5] E W Montroll and G H Weiss ldquoRandom walks on lattices IIrdquoJournal of Mathematical Physics vol 6 pp 167ndash181 1965
[6] H C Fogedby ldquoLangevin equations for continuous time Levyflightsrdquo Physical Review E vol 50 no 2 pp 1657ndash1660 1994
[7] S Bochner ldquoDiffusion equation and stochastic processesrdquoProceedings of the National Academy of Sciences vol 35 no 7pp 368ndash370 1949
[8] A Baule and R Friedrich ldquoJoint probability distributions for aclass of non-Markovian processesrdquo Physical Review E vol 71no 2 Article ID 026101 2005
[9] A Piryatinska A I Saichev and W A Woyczynski ldquoModelsof anomalous diffusion the subdiffusive caserdquo Physica AStatistical Mechanics and Its Applications vol 349 no 3-4 pp375ndash420 2005
[10] M Magdziarz A Weron and K Weron ldquoFractional Fokker-Planck dynamics stochastic representation and computer sim-ulationrdquo Physical Review EmdashStatistical Nonlinear and SoftMatter Physics vol 75 no 1 Article ID 016708 2007
[11] D Kleinhans and R Friedrich ldquoContinuous-time randomwalks simulation of continuous trajectoriesrdquo Physical Review Evol 76 no 6 Article ID 061102 2007
[12] AWeronMMagdziarz and KWeron ldquoModeling of subdiffu-sion in space-time-dependent force fields beyond the fractionalFokker-Planck equationrdquo Physical Review E vol 77 no 3Article ID 036704 2008
[13] M Magdziarz A Weron and J Klafter ldquoEquivalence of thefractional fokker-planck and subordinated langevin equationsthe case of a time-dependent forcerdquo Physical Review Letters vol101 no 21 Article ID 210601 2008
[14] A Weron and S Orzel ldquoIto formula for subordinated Langevinequation A case of time dependent forcerdquoActa Physica PolonicaB vol 40 no 5 pp 1271ndash1277 2009
[15] S Eule and R Friedrich ldquoSubordinated Langevin equationsfor anomalous diffusion in external potentialsmdashbiasing anddecoupled external forcesrdquo EPL vol 86 no 3 Article ID 300082009
[16] M Magdziarz ldquoLangevin picture of subdiffusion with infinitelydivisiblewaiting timesrdquo Journal of Statistical Physics vol 135 no4 pp 763ndash772 2009
[17] M Magdziarz ldquoStochastic representation of subdiffusion pro-cesses with time-dependent driftrdquo Stochastic Processes andTheirApplications vol 119 no 10 pp 3238ndash3252 2009
[18] M Magdziarz and A Weron ldquoCompetition between subdiffu-sion and Levy flights a Monte Carlo approachrdquo Physical ReviewE vol 75 Article ID 056702 2007
[19] B o Dybiec and E Gudowska-Nowak ldquoSubordinated diffusionand continuous time random walk asymptoticsrdquo Chaos vol 20no 4 Article ID 043129 2010
[20] A Wyłomanska ldquoArithmetic Brownian motion subordinatedby tempered stable and inverse tempered stable processesrdquoPhysica A StatisticalMechanics and Its Applications vol 391 no22 pp 5685ndash5696 2012
[21] H Gu J-R Liang and Y-X Zhang ldquoOn a time-changedgeometric Brownian motion and its application in financialmarketrdquo Acta Physica Polonica B vol 43 no 8 pp 1667ndash16812012
[22] J Gajda and A Wyłomanska ldquoGeometric Brownian motionwith tempered stable waiting timesrdquo Journal of StatisticalPhysics vol 148 no 2 pp 296ndash305 2012
[23] J Janczura S Orzeł and A Wyłomanska ldquoSubordinated 120572120572-stable Ornstein-Uhlenbeck process as a tool for financial datadescriptionrdquo Physica A Statistical Mechanics and its Applica-tions vol 390 no 23-24 pp 4379ndash4387 2011
[24] J Gajda and AWyłomanska ldquoTime-changedOrnsteinndashUhlen-beck processrdquo Journal of Physics A Mathematical and Theoreti-cal vol 48 no 13 Article ID 135004 2015
[25] A Wyłomanska ldquoThe tempered stable process with infinitelydivisible inverse subordinatorsrdquo Journal of StatisticalMechanicsTheory and Experiment vol 2013 no 10 Article ID P10011 2013
[26] Y-X Zhang H Gu and J-R Liang ldquoFokker-planck typeequations associated with subordinated processes controlled bytempered 120572-stable processesrdquo Journal of Statistical Physics vol152 no 4 pp 742ndash752 2013
[27] J Gajda and A Wyłomanska ldquoFokkerndashPlanck type equationsassociated with fractional Brownian motion controlled byinfinitely divisible processesrdquo Physica A Statistical Mechanicsand Its Applications vol 405 pp 104ndash113 2014
[28] M Teuerle A Wyłomanska and G Sikora ldquoModeling anoma-lous diffusion by a subordinated fractional Levy-stable processrdquoJournal of Statistical Mechanics Theory and Experiment vol2013 no 5 Article ID P05016 2013
[29] J Gajda andMMagdziarz ldquoFractional Fokker-Planck equationwith tempered 120572-stable waiting times Langevin picture andcomputer simulationrdquo Physical Review E Statistical Nonlinearand Soft Matter Physics vol 82 no 1 Article ID 011117 2010
[30] J Janczura and A Wyłomanska ldquoAnomalous diffusion mod-els different types of subordinator distributionrdquo Acta PhysicaPolonica B vol 43 no 5 pp 1001ndash1016 2012
[31] J Gajda ldquoFractional FokkerndashPlanck equationwith space depen-dent drift and diffusion the case of tempered 120572-stable waiting-timesrdquo Jagellonian University Institute of Physics Acta PhysicaPolonica B vol 44 no 5 pp 1149ndash1161 2013
[32] C Song T Koren P Wang and A Barabasi ldquoModelling thescaling properties of humanmobilityrdquoNature Physics vol 6 no10 pp 818ndash823 2010
[33] A Maye C-H Hsieh G Sugihara and B Brembs ldquoOrder inspontaneous behaviorrdquo PLoS ONE vol 2 no 5 article e4432007
[34] E Scalas ldquoThe application of continuous-time randomwalks infinance and economicsrdquo Physica A Statistical Mechanics and itsApplications vol 362 no 2 pp 225ndash239 2006
[35] A Chechkin M Hofmann and I M Sokolov ldquoContinuous-time random walk with correlated waiting timesrdquo PhysicalReview E Statistical Nonlinear and Soft Matter Physics vol 80no 3 Article ID 031112 2009
[36] M M Meerschaert E Nane and Y Xiao ldquoCorrelated continu-ous time random walksrdquo Statistics amp Probability Letters vol 79no 9 pp 1194ndash1202 2009
Advances in Mathematical Physics 7
[37] V Tejedor and R Metzler ldquoAnomalous diffusion in correlatedcontinuous time random walksrdquo Journal of Physics A Mathe-matical and Theoretical vol 43 no 8 Article ID 082002 2010
[38] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E vol 85 no 5 Article ID 051103 2012
[39] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time randomwalksmdashscaling limits andLangevin picturerdquo Journal of Statistical Mechanics Theory andExperiment vol 2012 no 4 Article ID P04010 2012
[40] M Magdziarz W Szczotka and P Zebrowski ldquoAsymptoticbehaviour of randomwalks with correlated temporal structurerdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 469 no 2159 Article ID 201304192013
[41] J H Schulz A V Chechkin and R Metzler ldquoCorrelatedcontinuous time random walks combining scale-invariancewith long-range memory for spatial and temporal dynamicsrdquoJournal of Physics A Mathematical and Theoretical vol 46 no47 Article ID 475001 2013
[42] L Shi Z Yu H Huang ZMao andA Xiao ldquoThe subordinatedprocesses controlled by a family of subordinators and corre-sponding FokkerndashPlanck type equationsrdquo Journal of StatisticalMechanics Theory and Experiment vol 2014 no 12 Article IDP12002 2014
[43] J Wang J Zhou L Lv W Qiu and F Ren ldquoHeterogeneousmemorized continuous time random walks in an external forcefieldsrdquo Journal of Statistical Physics vol 156 no 6 pp 1111ndash11242014
[44] F-Y Ren J Wang L-J Lv H Pan and W-Y Qiu ldquoEffect ofdifferent waiting time processes with memory to anomalousdiffusion dynamics in an external force fieldsrdquo Physica AStatistical Mechanics and Its Applications vol 417 pp 202ndash2142015
[45] L Lv F-Y Ren J Wang and J Xiao ldquoCorrelated continuoustime random walk with time averaged waiting timerdquo Physica AStatistical Mechanics and Its Applications vol 422 pp 101ndash1062015
[46] T W Burkhardt ldquoSemiflexible polymer in the half plane andstatistics of the integral of a Brownian curverdquo Journal of PhysicsA Mathematical and General vol 26 no 22 pp L1157ndashL11621993
[47] P Valageas ldquoStatistical properties of the burgers equation withbrownian initial velocityrdquo Journal of Statistical Physics vol 134no 3 pp 589ndash640 2009
[48] TW Burkhardt ldquoThe random acceleration process in boundedgeometriesrdquo Journal of Statistical Mechanics Theory and Exper-iment vol 2007 no 7 Article ID P07004 2007
[49] J Jeon A V Chechkin and R Metzler ldquoScaled Brownianmotion a paradoxical process with a time dependent diffusivityfor the description of anomalous diffusionrdquo Physical ChemistryChemical Physics vol 16 no 30 pp 15811ndash15817 2014
[50] Y He S Burov R Metzler and E Barkai ldquoRandom time-scaleinvariant diffusion and transport coefficientsrdquo Physical ReviewLetters vol 101 no 5 Article ID 058101 2008
[51] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 2009
[52] J-H Jeon and R Metzler ldquoFractional Brownian motion andmotion governed by the fractional Langevin equation in con-fined geometriesrdquo Physical Review E vol 81 no 2 Article ID021103 2010
[53] E Barkai Y Garini and R Metzler ldquoStrange kinetics of singlemolecules in living cellsrdquo Physics Today vol 65 no 8 pp 29ndash352012
[54] J Kursawe J Schulz and R Metzler ldquoTransient aging infractional Brownian and Langevin-equation motionrdquo PhysicalReview E vol 88 no 6 Article ID 062124 2013
[55] R Metzler J-H Jeon A G Cherstvy and E Barkai ldquoAnoma-lous diffusion models and their properties non-stationaritynon-ergodicity and ageing at the centenary of single particletrackingrdquo Physical Chemistry Chemical Physics vol 16 no 44pp 24128ndash24164 2014
[56] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
[37] V Tejedor and R Metzler ldquoAnomalous diffusion in correlatedcontinuous time random walksrdquo Journal of Physics A Mathe-matical and Theoretical vol 43 no 8 Article ID 082002 2010
[38] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time random walks in external forcefieldsrdquo Physical Review E vol 85 no 5 Article ID 051103 2012
[39] M Magdziarz R Metzler W Szczotka and P ZebrowskildquoCorrelated continuous-time randomwalksmdashscaling limits andLangevin picturerdquo Journal of Statistical Mechanics Theory andExperiment vol 2012 no 4 Article ID P04010 2012
[40] M Magdziarz W Szczotka and P Zebrowski ldquoAsymptoticbehaviour of randomwalks with correlated temporal structurerdquoProceedings of the Royal Society A Mathematical Physical andEngineering Sciences vol 469 no 2159 Article ID 201304192013
[41] J H Schulz A V Chechkin and R Metzler ldquoCorrelatedcontinuous time random walks combining scale-invariancewith long-range memory for spatial and temporal dynamicsrdquoJournal of Physics A Mathematical and Theoretical vol 46 no47 Article ID 475001 2013
[42] L Shi Z Yu H Huang ZMao andA Xiao ldquoThe subordinatedprocesses controlled by a family of subordinators and corre-sponding FokkerndashPlanck type equationsrdquo Journal of StatisticalMechanics Theory and Experiment vol 2014 no 12 Article IDP12002 2014
[43] J Wang J Zhou L Lv W Qiu and F Ren ldquoHeterogeneousmemorized continuous time random walks in an external forcefieldsrdquo Journal of Statistical Physics vol 156 no 6 pp 1111ndash11242014
[44] F-Y Ren J Wang L-J Lv H Pan and W-Y Qiu ldquoEffect ofdifferent waiting time processes with memory to anomalousdiffusion dynamics in an external force fieldsrdquo Physica AStatistical Mechanics and Its Applications vol 417 pp 202ndash2142015
[45] L Lv F-Y Ren J Wang and J Xiao ldquoCorrelated continuoustime random walk with time averaged waiting timerdquo Physica AStatistical Mechanics and Its Applications vol 422 pp 101ndash1062015
[46] T W Burkhardt ldquoSemiflexible polymer in the half plane andstatistics of the integral of a Brownian curverdquo Journal of PhysicsA Mathematical and General vol 26 no 22 pp L1157ndashL11621993
[47] P Valageas ldquoStatistical properties of the burgers equation withbrownian initial velocityrdquo Journal of Statistical Physics vol 134no 3 pp 589ndash640 2009
[48] TW Burkhardt ldquoThe random acceleration process in boundedgeometriesrdquo Journal of Statistical Mechanics Theory and Exper-iment vol 2007 no 7 Article ID P07004 2007
[49] J Jeon A V Chechkin and R Metzler ldquoScaled Brownianmotion a paradoxical process with a time dependent diffusivityfor the description of anomalous diffusionrdquo Physical ChemistryChemical Physics vol 16 no 30 pp 15811ndash15817 2014
[50] Y He S Burov R Metzler and E Barkai ldquoRandom time-scaleinvariant diffusion and transport coefficientsrdquo Physical ReviewLetters vol 101 no 5 Article ID 058101 2008
[51] W Deng and E Barkai ldquoErgodic properties of fractionalBrownian-Langevin motionrdquo Physical Review E vol 79 no 1Article ID 011112 2009
[52] J-H Jeon and R Metzler ldquoFractional Brownian motion andmotion governed by the fractional Langevin equation in con-fined geometriesrdquo Physical Review E vol 81 no 2 Article ID021103 2010
[53] E Barkai Y Garini and R Metzler ldquoStrange kinetics of singlemolecules in living cellsrdquo Physics Today vol 65 no 8 pp 29ndash352012
[54] J Kursawe J Schulz and R Metzler ldquoTransient aging infractional Brownian and Langevin-equation motionrdquo PhysicalReview E vol 88 no 6 Article ID 062124 2013
[55] R Metzler J-H Jeon A G Cherstvy and E Barkai ldquoAnoma-lous diffusion models and their properties non-stationaritynon-ergodicity and ageing at the centenary of single particletrackingrdquo Physical Chemistry Chemical Physics vol 16 no 44pp 24128ndash24164 2014
[56] I Podlubny Fractional Differential Equations Academic PressSan Diego Calif USA 1999
![Question 1 · 2020-05-28 · 1 Question 1 a) RTS: 1) '[ 𝑡| 0= 0]= 𝑡 0 2) '[ 𝑡| 0= 0]= 𝑡−1(1− ) 0 In the Greenwood model. We are given that '[ 𝑡| 𝑡−1]= 𝑡](https://static.documents.pub/doc/80x56/5f04111d7e708231d40c2788/question-1-2020-05-28-1-question-1-a-rts-1-0-0-0-2-.jpg)
![HOW TO SUM FREQUENCY AND SECOND HARMONIC ......THEORY 𝑃 𝑡=𝜖 0[χ1𝐸 𝑡+χ2𝐸 𝑡+χ3𝐸 𝑡+⋯] 2ND ORDER NONLINEAR SUSCEPTIBILITY • SFG/ SHG • SELECTIVITY AT](https://static.documents.pub/doc/80x56/60ac142e19adaa1130452584/how-to-sum-frequency-and-second-harmonic-theory-f-oe-01.jpg)
![cobertura global y de orden superior · Black-Scholes [1973], Merton [1973] 1. por el lema de Itō, si =𝜇 𝑡+𝜎 , entonces 2. construya el portafolio Π𝑡= 𝑡+𝛿 𝑡,](https://static.documents.pub/doc/80x56/5ec06b64d0fc0738f0216a9e/cobertura-global-y-de-orden-superior-black-scholes-1973-merton-1973-1-por.jpg)
