Functionals of fractional Brownian motion and the three arcsine laws

被引:8
作者
Sadhu, Tridib [1 ]
Wiese, Kay Jorg [2 ]
机构
[1] Tata Inst Fundamental Res, Dept Theoret Phys, Dr Homi Bhabha Rd, Mumbai 400005, Maharashtra, India
[2] Univ Paris Diderot, Sorbonne Univ, Sorbonne Paris Cite, Lab Phys,Ecole Normale Super,CNRS, 24 Rue Lhomond, F-75005 Paris, France
关键词
SINGLE-FILE DIFFUSION; ANOMALOUS DIFFUSION; RANDOM-WALKS; SINE LAW; TIME; STATISTICS; DYNAMICS; ANALOGS; MAXIMUM; PHYSICS;
D O I
10.1103/PhysRevE.104.054112
中图分类号
O35 [流体力学]; O53 [等离子体物理学];
学科分类号
070204 ; 080103 ; 080704 ;
摘要
Fractional Brownian motion is a non-Markovian Gaussian process indexed by the Hurst exponent H is an element of (0, 1), generalizing standard Brownian motion to account for anomalous diffusion. Functionals of this process are important for practical applications as a standard reference point for nonequilibrium dynamics. We describe a perturbation expansion allowing us to evaluate many nontrivial observables analytically: We generalize the celebrated three arcsine laws of standard Brownian motion. The functionals are: (i) the fraction of time the process remains positive, (ii) the time when the process last visits the origin, and (iii) the time when it achieves its maximum (or minimum). We derive expressions for the probability of these three functionals as an expansion in epsilon = H - 12, up to second order. We find that the three probabilities are different, except for H = 12, where they coincide. Our results are confirmed to high precision by numerical simulations.
引用
收藏
页数:47
相关论文
共 79 条
[1]  
[Anonymous], 2010, CAMBRIDGE SERIES STA
[2]  
[Anonymous], 2004, FINANCIAL DERIVATIVE
[3]   Extreme events for fractional Brownian motion with drift: Theory and numerical validation [J].
Arutkin, Maxence ;
Walter, Benjamin ;
Wiese, Kay Jorg .
PHYSICAL REVIEW E, 2020, 102 (02)
[4]  
BARLOW M, 1989, LECT NOTES MATH, V1372, P294
[5]  
Biagini F, 2008, PROBAB APPL SER, P1
[6]   ON HIGHER-DIMENSIONAL ANALOGS OF THE ARC-SINE LAW [J].
BINGHAM, NH ;
DONEY, RA .
JOURNAL OF APPLIED PROBABILITY, 1988, 25 (01) :120-131
[7]   ANOMALOUS DIFFUSION IN DISORDERED MEDIA - STATISTICAL MECHANISMS, MODELS AND PHYSICAL APPLICATIONS [J].
BOUCHAUD, JP ;
GEORGES, A .
PHYSICS REPORTS-REVIEW SECTION OF PHYSICS LETTERS, 1990, 195 (4-5) :127-293
[8]   Universal Algorithm for Identification of Fractional Brownian Motion. A Case of Telomere Subdiffusion [J].
Burnecki, Krzysztof ;
Kepten, Eldad ;
Janczura, Joanna ;
Bronshtein, Irena ;
Garini, Yuval ;
Weron, Aleksander .
BIOPHYSICAL JOURNAL, 2012, 103 (09) :1839-1847
[9]   On Distributions of Functionals of Anomalous Diffusion Paths [J].
Carmi, Shai ;
Turgeman, Lior ;
Barkai, Eli .
JOURNAL OF STATISTICAL PHYSICS, 2010, 141 (06) :1071-1092
[10]   SOME EXTENSIONS OF THE ARE SINE LAW AS PARTIAL CONSEQUENCES OF THE SCALING PROPERTY OF BROWNIAN-MOTION [J].
CARMONA, P ;
PETIT, F ;
YOR, M .
PROBABILITY THEORY AND RELATED FIELDS, 1994, 100 (01) :1-29