Mouvement brownien fractionnaire

Le mouvement brownien fractionnaire (mBf) a été introduit par Kolmogorov en 1940, comme moyen d'engendrer des "spirales" gaussiennes dans des espaces de Hilbert. En 1968, Mandelbrot et Van Ness l'ont rendu célèbre en l'introduisant dans des modèles financiers, et en étudiant ses propriétés. Le champ des applications du mBf est immense. En effet, il sert par exemple à recréer certains paysages naturels, notamment des montagnes, mais également en hydrologie, télécommunications, économie, physique... Rudiments mathématiques Définition du mBf Le mouvement brownien fractionnaire d'exposant de Hurst \alpha \in (0,1), noté {B_{\alpha}(t)}_{t\in \mathbb{R}}, est l'unique processus gaussien centré, nul en zéro et continu, dont la covariance est donnée par : \mathbb{E} (B_{\alpha}(s)B_{\alpha}(t)) = \frac{C_{\alpha}}{2}(|s|^{2\alpha}+|t|^{2\alpha}-|s-t|^{2\alpha}), où C_{\alpha} = \rm{Var}(B_{\alpha}(1))

Propriétés fines des trajectoires du mouvement brownien fractionnaire

Driss Baraka

Let X = {X(t); t ∈ RN} be a (N,d) fractional Brownian motion in Rd of index H ∈ (0,1). We study the local time of X for all temporal dimensions N and spatial dimensions d for which local time exist. We obtain two main results : R1. If we denote by Lx(I) the local time of X at x on I ⊂ RN, then there exists a positive finite constant c such that mφ(X-1(0) ∩ [0,1]N) = c L0([0,1]N), where φ(r) = rN-dH (log log 1/r)dH/N and mφ(E) is the Hausdorff φ-measure of E. This solves the problem of the relationship between the local time and the exact Hausdorff measure of zero set for X. R2. We refine results of Xiao (1997) for the local times of (N,d) fractional Brownian motion. We prove the law of iterated logarithm and global Hölder condition for the local time for our process. These results establish interesting properties which were only partially proved in the literature. This literature began with the work of Taylor and Wendel (1966) and Perkins (1981) for the first result; Kesten (1965) and Perkins (1981) for the second on the Brownian motion. It continued with several works of which that of Xiao (1997) on the locally nondeterministic processes with stationnary increments including the (N,d) fractional Brownian motion. An intermediate result is found to solve the case N > 1. We generalize the result of Kasahara et al. (1999) on the tail probability of local time.

EPFL2008

Development of statistical analysis tools for global observable of fluctuation data towards optimised real-time control applications

Garance Hélène Salomé Durr-Legoupil-Nicoud

The StatComp package is a Matlab statistical toolbox developed over the years by Dr. Testa and his students. It has been inspired by M. R. Brown’s paper Magnetohydrodynamic Turbulence: Observation and experiment [2]. It first performed the analysis of the edge magnetic turbulent field in the TCV. It started in 2015 by A. Yantchenko and has been constantly improved and supplemented since then. The last addition to the package was many separate functions for the ”big data” analysis of the results, done by S. Ogier-Collin. The entire code is currently under review for release in the MHD analysis package within the SPC’s General Analysis Toolkit. The present document reports the latest evolution of this package in the perspective of using the charac- terisation plasma turbulence to possibly provide useful information for the optimisation of real-time plasma control and the fusion performance of a tokamak. The mathematical theory of the StatComp analyses and some examples of application are presented in the section 2. The section 3 presents the evolution of the existing functions as well as the addition of the loading function for the electrostatic data from the edge of the plasma, and the multifractality and predictability analyses. These enhancements are put in the perspective of one particular usage: the characterisation of the turbulence in order optimise potentially plasma control. Then, the up-to-date running instructions and interpretation guidelines are detailed in the section 4. The latter are based on the output figures resulting of the analysis of a standard dataset constituted of a white noise sample, three fractional Brownian motions of different known Hurst index, of a linear ramp and of a sample of the solar wind. The section 5 shows the results of the test on four actual shots realised on the TCV tokamak. The varying parameters are the signs of the poloidal magnetic field and of the plasma current. The four shots are each the resultant of a positive or negative poloidal field and a positive or negative plasma current. The shape and position of the plasma in the vacuum vessel are the same for each shot as well as the amplitude of the varied parameters, i.e. the magnetic field and plasma current. The emphasis is made on the presentation and interpretation of the results obtained with the electrostatic data on the low-field side of the plasma. The obtained results are discussed along the limits of the package and its possible improvements in section 6 before concluding in section 7. In the appendix, the structures necessary to the use of the package are detailed and examples of run commands are presented. In order to offer to the reader a frame of reference for reflection, the main parameters and orders of magnitude related to the plasma shots in TCV are given. Some of the mathematical basis of the statistical theory are also elaborated to complete the description of the different tools of the package. Finally, the reduced bibliography of all the sources explicitly mentioned in this report is doubled by a second bibliography presenting a wider selection of relevant sources each accompanied with a brief description of its content and its link to the present study.

2022

Development of statistical analysis tools for global observable of fluctuation data towards optimised real-time control applications

Garance Hélène Salomé Durr-Legoupil-Nicoud

2022

Mouvement brownien fractionnaire

Propriétés fines des trajectoires du mouvement brownien fractionnaire

Development of statistical analysis tools for global observable of fluctuation data towards optimised real-time control applications

The Exact Hausdorff Measure of the Zero Set of Fractional Brownian Motion

Propriétés fines des trajectoires du mouvement brownien fractionnaire

Development of statistical analysis tools for global observable of fluctuation data towards optimised real-time control applications

The Exact Hausdorff Measure of the Zero Set of Fractional Brownian Motion