The geometry of the space of branched rough paths

We construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between theses two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker-Campbell-Hausdorff formula, on a constructive version of the Lyons-Victoir extension theorem and on the Hairer-Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths

The geometry of the space of branched rough paths

We construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between theses two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker-Campbell-Hausdorff formula, on a constructive version of the Lyons-Victoir extension theorem and on the Hairer-Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths

The Nagaev-Guivarc'h method via the Keller-Liverani theorem

The Nagaev-Guivarc'h method, via the perturbation operator theorem of Keller and Liverani, has been exploited in recent papers to establish local limit and Berry-Essen type theorems for unbounded functionals of strongly ergodic Markov chains. The main difficulty of this approach is to prove Taylor expansions for the dominating eigenvalue of the Fourier kernels. This paper outlines this method and extends it by proving a multi-dimensional local limit theorem, a first-order Edgeworth expansion, and a multi-dimensional Berry-Esseen type theorem in the sense of Prohorov metric. When applied to uniformly or geometrically ergodic chains and to iterative Lipschitz models, the above cited limit theorems hold under moment conditions similar, or close, to those of the i.i.d. case

Non-Fock Ground States in the Translation-Invariant Nelson Model Revisited Non-Perturbatively

The Nelson model, describing a quantum mechanical particle linearly coupled to a bosonic field, exhibits the infrared problem in the sense that no ground state exists at arbitrary total momentum. However, passing to a non-Fock representation, one can prove the existence of so-called dressed one-particle states. In this article, we give a simple non-perturbative proof for the existence of such one-particle states at arbitrary coupling strength and for almost all total momenta in a physically motivated momentum region. Our results hold both for the non- and the semi-relativistic Nelson model.Comment: 34 page

Stationary Kernels and Gaussian Processes on Lie Groups and their Homogeneous Spaces II: non-compact symmetric spaces

Gaussian processes are arguably the most important class of spatiotemporal models within machine learning. They encode prior information about the modeled function and can be used for exact or approximate Bayesian learning. In many applications, particularly in physical sciences and engineering, but also in areas such as geostatistics and neuroscience, invariance to symmetries is one of the most fundamental forms of prior information one can consider. The invariance of a Gaussian process' covariance to such symmetries gives rise to the most natural generalization of the concept of stationarity to such spaces. In this work, we develop constructive and practical techniques for building stationary Gaussian processes on a very large class of non-Euclidean spaces arising in the context of symmetries. Our techniques make it possible to (i) calculate covariance kernels and (ii) sample from prior and posterior Gaussian processes defined on such spaces, both in a practical manner. This work is split into two parts, each involving different technical considerations: part I studies compact spaces, while part II studies non-compact spaces possessing certain structure. Our contributions make the non-Euclidean Gaussian process models we study compatible with well-understood computational techniques available in standard Gaussian process software packages, thereby making them accessible to practitioners

Dual weights in the theory of harmonic Siegel modular forms

We define harmonic Siegel modular forms based on a completely new approach using vector-valued covariant operators. The Fourier expansions of such forms are investigated for two distinct slash actions. Two very different reasons are given why these slash actions are natural. We prove that they are related by xi-operators that generalize the xi-operator for elliptic modular forms. We call them dual slash actions or dual weights, a name which is suggested by the many properties that parallel the elliptic case. Based on Kohnen's limit process for real-analytic Siegel Eisenstein series, we show that, under mild assumptions, Jacobi forms can be obtained from harmonic Siegel modular forms, generalizing the classical Fourier-Jacobi expansion. The resulting Fourier-Jacobi coefficients are harmonic Maass-Jacobi forms, which are defined in full generality in this work. A compatibility between the various xi-operators for Siegel modular forms, Jacobi forms, and elliptic modular forms is deduced, relating all three kinds of modular forms.Duale Gewichte in der Theorie harmonischer Siegelscher Modulformen Fußend auf einem vollständig neuen Ansatz, dem vektorwertige kovariante Operatoren zu Grunde liegen, definieren wir den Begriff der harmonischen Siegelschen Modulform. Dieser Definition schließt sich eine Untersuchung der für zwei verschiedene Strichoperationen auftretenden Fourier-Entwicklungen an. Die besagten Operationen sind natürlich in zweierlei Hinsicht, auf die wir beide näher eingehen. Darüber hinaus besteht eine Verbindung zwischen diesen beide Strichoperatoren, die durch zwei xi-Operatoren, die wiederum den elliptischen xi-Operator verallgemeinern, vermittelt wird. Die bemerkenswerte Ähnlichkeit zum Verhalten von elliptischen Modulformen dual Gewichts legt die Verwendung dieses Begriffs auch für die hier untersuchten Gewichte Siegelscher Modulformen nahe. Eine Verallgemeinerung der klassischen Fourier-Jacobi-Entwicklung kann aufbauend auf Kohnens Grenzwertprozess für reell-analytische Siegelsche Eisensteinreihen für eine große Klasse von harmonischen Siegelschen Modulformen hergele\-tet werden. Die herbei auftretenden Fourier-Jacobi-Entwicklungen stellen sich als Maaß-Jacobiformen heraus, die in voller Allgemeinheit in dieser Arbeit definiert werden. Wir zeigen schließlich, dass die verschiedenen xi-Operatoren für Siegelsche Modulformen, Jacobiformen und elliptische Modulformen miteinander verträglich sind und stellen so einen Zusammenhang zwischen diesen drei Arten von Modulformen her