Fractional Brownian bridges

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Table of contents

1 Introduction 
1.1 Statistical physics and disordered systems
1.2 Probability theory and stochastic processes
1.2.1 Random walks and Brownian motion
1.2.2 Markov processes
1.2.3 The three Levy’s arcsine laws
1.2.4 Anomalous diffusion and self-similarity
1.2.5 Extreme-value statistics and persistence
1.3 Fractional Brownian motion (fBm)
1.3.1 Definition and properties
1.3.2 History and applications
1.3.3 Numerical simulations
1.3.4 Fractional Brownian bridges
1.3.5 Pickands constants
1.4 Elastic interfaces in disordered media
1.4.1 Generical ideas
1.4.2 Some applications and experimental realisations
1.4.3 Functional renormalisation
1.4.4 The mean field model and the Brownian force model
2 Extreme-value statistics of fractional Brownian motion 
2.1 Presentation of the chapter
2.2 Perturbative approach to fBm
2.2.1 Path integral formulation and the action
2.2.2 The order-0 term
2.2.3 The first-order terms
2.2.4 Graphical representation and diagrams
2.3 Analytical Results
2.3.1 Scaling results
2.3.2 The complete result
2.3.3 The third arcsine law
2.3.4 The distribution of the maximum
2.3.5 Survival probability
2.3.6 Joint distribution
2.4 Numerical Results
2.4.1 The third arcsine Law
2.4.2 The distribution of the maximum
2.5 Conclusions
2.A Details on the perturbative expansion
2.B Recall of an old result
2.C Computation of the new correction
2.D Correction to the third arcsine Law
2.E Correction to the maximum-value distribution
2.F Correction to the survival distribution
2.G Special functions and some inverse Laplace transforms
2.H Check of the covariance function
2.I The Davis and Harte algorithm
3 The first and second arcsine laws 
3.1 Presentation of the chapter
3.2 Positive time of a Brownian motion
3.2.1 Positive time of a discrete random walk
3.2.2 Propagators in continuous time
3.3 Time of a fBm remains positive
3.4 Last zero of a fBm
3.4.1 The Brownian case
3.4.2 Scaling and perturbative expansion for the distribution of the last zero
3.4.3 The two non-trivial diagrams at second order
3.A Action at second order
4 Fractional Brownian bridges and positive time 
4.1 Presentation of the chapter
4.2 Preliminaries: Gaussian Bridges
4.3 Time a fBm birdge remains positive
4.3.1 Scale invariance and a useful transformation
4.3.2 FBm Bridge
4.3.3 Numerical results
4.4 Extremum of fBm Bridges
4.4.1 Distribution of the time to reach the maximum
4.4.2 The maximum-value distribution
4.4.3 Optimal path for fBm, and the tail of the maximum distribution
4.4.4 Joint Distribution
4.5 Conclusions
4.A Details on correlation functions for Gaussian bridges
4.B Abel transform
4.C Inverse Laplace transforms, and other useful relations
5 FBm with drift and Pickands constants 
5.1 Presentation of the chapter
5.2 Brownian motion with drift
5.3 Pertubative expansion around Brownian motion
5.3.1 Action with drift
5.3.2 Survival probability and Pickands constants
5.3.3 Maximum-value distribution in the large time limit
5.4 Maximum-value of a fBm Bridge and Pickands constants
5.A Derivation of the action
5.B Details of calculations
5.C Scaling
6 Avalanches in the Brownian force model 
6.1 Presentation of the chapter
6.2 Avalanche observables in the BFM
6.2.1 The Brownian Force Model
6.2.2 Avalanche observables and scaling
6.2.3 Generating functions and instanton equation
6.3 Distribution of avalanche size
6.3.1 Global size
6.3.2 Local size
6.3.3 Joint global and local size
6.3.4 Scaling exponents
6.4 Driving at a point: avalanche sizes
6.4.1 Imposed local force
6.4.2 Imposed displacement at a point
6.5 Distribution of avalanche extension
6.5.1 Scaling arguments for the distribution of extension
6.5.2 Instanton equation for two local sizes
6.5.3 Avalanche extension with a local kick
6.5.4 Avalanche extension with a uniform kick
6.6 Non-stationnary dynamics in the BFM
6.7 Conclusion
6.A Airy functions
6.B General considerations on the instanton equation
6.C Distribution of local size
6.D Joint density
6.E Imposed local displacement
6.F Some elliptic integrals for the distribution of avalanche extension
6.G Joint distribution for extension and total size
6.H Numerics
6.I Weierstrass and Elliptic functions
6.J Non-stationary dynamics
7 General conclusion
Bibliographie