Unified View on L\'evy White Noises: General Integrability Conditions and Applications to Linear SPDE

There exists several ways of constructing L\'evy white noise, for instance are as a generalized random process in the sense of I.M. Gelfand and N.Y. Vilenkin, or as an independently scattered random measure introduced by B.S. Rajput and J. Rosinski. In this article, we unify those two approaches by extending the L\'evy white noise, defined as a generalized random process, to an independently scattered random measure. We are then able to give general integrability conditions for L\'evy white noises, thereby maximally extending their domain of definition. Based on this connection, we provide new criteria for the practical determination of this domain of definition, including specific results for the subfamilies of Gaussian, symmetric-$\alpha$-stable, Laplace, and compound Poisson noises. We also apply our results to formulate a general criterion for the existence of generalized solutions of linear stochastic partial differential equations driven by a L\'evy white noise.Comment: 43 page

Periodic Splines and Gaussian Processes for the Resolution of Linear Inverse Problems

This paper deals with the resolution of inverse problems in a periodic setting or, in other terms, the reconstruction of periodic continuous-domain signals from their noisy measurements. We focus on two reconstruction paradigms: variational and statistical. In the variational approach, the reconstructed signal is solution to an optimization problem that establishes a tradeoff between fidelity to the data and smoothness conditions via a quadratic regularization associated to a linear operator. In the statistical approach, the signal is modeled as a stationary random process defined from a Gaussian white noise and a whitening operator; one then looks for the optimal estimator in the mean-square sense. We give a generic form of the reconstructed signals for both approaches, allowing for a rigorous comparison of the two.We fully characterize the conditions under which the two formulations yield the same solution, which is a periodic spline in the case of sampling measurements. We also show that this equivalence between the two approaches remains valid on simulations for a broad class of problems. This extends the practical range of applicability of the variational method

On the Continuity of Characteristic Functionals and Sparse Stochastic Modeling

The characteristic functional is the infinite-dimensional generalization of the Fourier transform for measures on function spaces. It characterizes the statistical law of the associated stochastic process in the same way as a characteristic function specifies the probability distribution of its corresponding random variable. Our goal in this work is to lay the foundations of the innovation model, a (possibly) non-Gaussian probabilistic model for sparse signals. This is achieved by using the characteristic functional to specify sparse stochastic processes that are defined as linear transformations of general continuous-domain white noises (also called innovation processes). We prove the existence of a broad class of sparse processes by using the Minlos-Bochner theorem. This requires a careful study of the regularity properties, especially the boundedness in Lp-spaces, of the characteristic functional of the innovations. We are especially interested in the functionals that are only defined for p<1 since they appear to be associated with the sparser kind of processes. Finally, we apply our main theorem of existence to two specific subclasses of processes with specific invariance properties.Comment: 24 page

Gaussian versus Sparse Stochastic Processes:Construction, Regularity, Compressibility

Although our work lies in the field of random processes, this thesis was originally motivated by signal processing applications, mainly the stochastic modeling of sparse signals. We develop a mathematical study of the innovation model, under which a signal is described as a random process s that can be linearly and deterministically transformed into a white noise. The noise represents the unpredictable part of the signal, called its innovation, and is part of the family of Lévy white noises, which includes both Gaussian and Poisson noises. In mathematical terms, s satisfies the equation Ls=w where L is a differential operator and w a Lévy noise. The problem is therefore to study the solution of a stochastic differential equation driven by a Lévy noise. Gaussian models usually fail to reproduce the empirical sparsity observed in real-world signals. By contrast, Lévy models offer a wide range of random processes going from typically non-sparse (Gaussian) to very sparse ones (Poisson), and with many sparse signals standing between these two extremes. Our contributions can be divided in four parts. First, the cornerstone of our work is the theory of generalized random processes. Within this framework, all the considered random processes are seen as random tempered generalized functions and can be observed through smooth and rapidly decaying windows. This allows us to define the solutions of Ls=w, called generalized Lévy processes, in the most general setting. Then, we identify two limit phenomenons: the approximation of generalized Lévy processes by their Poisson counterparts, and the asymptotic behavior of generalized Lévy processes at coarse and fine scales. In the third part, we study the localization of Lévy noise in notorious function spaces (Hölder, Sobolev, Besov). As an application, characterize the local smoothness and the asymptotic growth rate of the Lévy noise. Finally, we quantify the local compressibility of the generalized Lévy processes, understood as a measure of the decreasing rate of their approximation error in an appropriate basis. From this last result, we provide a theoretical justification of the ability of the innovation model to represent sparse signals. The guiding principle of our research is the duality between the local and asymptotic properties of generalized Lévy processes. In particular, we highlight the relevant quantities, called the local and asymptotic indices, that allow quantifying the local regularity, the asymptotic growth rate, the limit behavior at coarse and fine scales, and the level of compressibility of the solutions of generalized Lévy processes