Suprathermal ion transport in turbulent magnetized plasmas

Suprathermal ions, which have an energy greater than the quasi-Maxwellian background plasma temperature, are present in many laboratory and astrophysical plasmas. In fusion devices, they are generated by the fusion reactions and auxiliary heating. Controlling their transport is essential for the success of future fusion devices that could provide a clean, safe and abundant source of electric power to our society. In space, suprathermal ions include energetic solar particles and cosmic rays. The understanding of the acceleration and transport mechanisms of these particles is still incomplete. Basic plasma devices allow detailed measurements that are not accessible in astrophysical and fusion plasmas, due to the difficulty to access the former and the high temperatures of the latter. The basic toroidal device TORPEX offers an easy access for diagnostics, well characterized plasma scenarios and validated numerical simulations of its turbulence dynamics, making it the ideal platform for the investigation of suprathermal ion transport. This Thesis presents three-dimensional measurements of a suprathermal ion beam injected in turbulent TORPEX plasmas. The combination of uniquely resolved measurements and first-principle numerical simulations reveals the general non-diffusive nature of the suprathermal ion transport. A precise characterization of their transport regime shows that, depending on their energies, suprathermal ions can experience either a superdiffusive transport or a subdiffusive transport in the same background turbulence. The transport character is determined by the interaction of the suprathermal ion orbits with the turbulent plasma structures, which in turn depends on the ratio between the ion energy and the background plasma temperature. Time-resolved measurements reveal a clear difference in the intermittency of suprathermal ions time-traces depending on the transport regime they experience. Conditionally averaged measurements uncover the influence of field elongated turbulent structures, referred to as blobs, on the suprathermal ion beam. A theoretical model extending the Brownian motion to include non-Gaussian (Lévy) statistics and long-range temporal correlation is developed. This model successfully describes the evolution of the radial particle density from the numerical simulations and provides information on the microscopic processes underlying the non-diffusive transport of suprathermal ions

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Innovations in Quantitative Risk Management

Quantitative Finance; Game Theory, Economics, Social and Behav. Sciences; Finance/Investment/Banking; Actuarial Science

Continuous and discrete properties of stochastic processes

This thesis considers the interplay between the continuous and discrete properties of random stochastic processes. It is shown that the special cases of the one-sided Lévy-stable distributions can be connected to the class of discrete-stable distributions through a doubly-stochastic Poisson transform. This facilitates the creation of a one-sided stable process for which the N-fold statistics can be factorised explicitly. The evolution of the probability density functions is found through a Fokker-Planck style equation which is of the integro-differential type and contains non-local effects which are different for those postulated for a symmetric-stable process, or indeed the Gaussian process. Using the same Poisson transform interrelationship, an exact method for generating discrete-stable variates is found. It has already been shown that discrete-stable distributions occur in the crossing statistics of continuous processes whose autocorrelation exhibits fractal properties. The statistical properties of a nonlinear filter analogue of a phase-screen model are calculated, and the level crossings of the intensity analysed. It is found that rather than being Poisson, the distribution of the number of crossings over a long integration time is either binomial or negative binomial, depending solely on the Fano factor. The asymptotic properties of the inter-event density of the process are found to be accurately approximated by a function of the Fano factor and the mean of the crossings alone

Continuous and discrete properties of stochastic processes

This thesis considers the interplay between the continuous and discrete properties of random stochastic processes. It is shown that the special cases of the one-sided Lévy-stable distributions can be connected to the class of discrete-stable distributions through a doubly-stochastic Poisson transform. This facilitates the creation of a one-sided stable process for which the N-fold statistics can be factorised explicitly. The evolution of the probability density functions is found through a Fokker-Planck style equation which is of the integro-differential type and contains non-local effects which are different for those postulated for a symmetric-stable process, or indeed the Gaussian process. Using the same Poisson transform interrelationship, an exact method for generating discrete-stable variates is found. It has already been shown that discrete-stable distributions occur in the crossing statistics of continuous processes whose autocorrelation exhibits fractal properties. The statistical properties of a nonlinear filter analogue of a phase-screen model are calculated, and the level crossings of the intensity analysed. It is found that rather than being Poisson, the distribution of the number of crossings over a long integration time is either binomial or negative binomial, depending solely on the Fano factor. The asymptotic properties of the inter-event density of the process are found to be accurately approximated by a function of the Fano factor and the mean of the crossings alone

Weak regularization by degenerate L\'evy noise and its applications

After a general introduction about the regularization by noise phenomenon in the degenerate setting, the first part of this PhD thesis focuses at establishing the Schauder estimates, a useful analytical tool to prove also the well-posedness of stochastic differential equations (SDEs), for two different classes of Kolmogorov equations under a weak H\"ormander-like condition, whose coefficients lie in suitable anisotropic H\"older spaces with multi-indices of regularity. The first class considers a nonlinear system controlled by a symmetric stable operator acting only on some components. Our method of proof relies on a perturbative approach based on forward parametrix expansions through Duhamel-type formulas. Due to the low regularizing properties given by the degenerate setting, we also exploit some controls on Besov norms, in order to deal with the non-linear perturbation. As an extension of the first one, we also present Schauder estimates associated with a degenerate Ornstein-Uhlenbeck operator driven by a larger class of stable-like operators, like the relativistic or the Lamperti stable one. Exploiting a backward parametrix approach, the second part of this work aims at establishing the weak well-posedness for a degenerate chain of SDEs driven by the same class of stable-like processes, under the assumptions of the minimal H\"older regularity on the coefficients. As a by-product of our method, we also present Krylov-type estimates of independent interest for the associated canonical process. Finally, we show through suitable counter-examples the existence of an (almost) sharp threshold on the regularity exponents ensuring the weak well-posedness for the SDE. In connection with some possible applications to kinetic dynamics with friction, we conclude by investigating the stability of second-order perturbations for degenerate Kolmogorov operators in Lp and H\"older norms.Comment: Doctoral thesis, Corrected some typos from the previous version. arXiv admin note: text overlap with arXiv:2107.0432

Stochastic timeseries analysis in electric power systems and paleo-climate data

In this thesis a data science study of elementary stochastic processes is laid, aided with the development of two numerical software programmes, applied to power-grid frequency studies and Dansgaard--Oeschger events in paleo-climate data. Power-grid frequency is a key measure in power grid studies. It comprises the balance of power in a power grid at any instance. In this thesis an elementary Markovian Langevin-like stochastic process is employed, extending from existent literature, to show the basic elements of power-grid frequency dynamics can be modelled in such manner. Through a data science study of power-grid frequency data, it is shown that fluctuations scale in an inverse square-root relation with their size, alike any other stochastic process, confirming previous theoretical results. A simple Ornstein--Uhlenbeck is offered as a surrogate model for power-grid frequency dynamics, with a versatile input of driving deterministic functions, showing not surprisingly that driven stochastic processes with Gaussian noise do not necessarily show a Gaussian distribution. A study of the correlations between recordings of power-grid frequency in the same power-grid system reveals they are correlated, but a theoretical understanding is yet to be developed. A super-diffusive relaxation of amplitude synchronisation is shown to exist in space in coupled power-grid systems, whereas a linear relation is evidenced for the emergence of phase synchronisation. Two Python software packages are designed, offering the possibility to extract conditional moments for Markovian stochastic processes of any dimension, with a particular application for Markovian jump-diffusion processes for one-dimensional timeseries. Lastly, a study of Dansgaard--Oeschger events in recordings of paleoclimate data under the purview of bivariate Markovian jump-diffusion processes is proposed, augmented by a semi-theoretical study of bivariate stochastic processes, offering an explanation for the discontinuous transitions in these events and showing the existence of deterministic couplings between the recordings of the dust concentration and a proxy for the atmospheric temperature