1,721,006 research outputs found
Norm inflation for a non-linear heat equation with Gaussian initial conditions
Full text linkWe consider a non-linear heat equation posed on the -dimensional torus, where is a polynomial of
degree at most and is a bilinear map that is not a total derivative. We
show that, if the initial condition is taken from a sequence of smooth
Gaussian fields with a specified covariance, then exhibits norm inflation
with high probability. A consequence of this result is that there exists no
Banach space of distributions which carries the Gaussian free field on the 3D
torus and to which the DeTurck-Yang-Mills heat flow extends continuously, which
complements recent well-posedness results in arXiv:2111.10652 and
arXiv:2201.03487. Another consequence is that the (deterministic) non-linear
heat equation exhibits norm inflation, and is thus locally ill-posed, at every
point in the Besov space ; the space
is an endpoint since the equation is locally
well-posed for for every .Comment: 21 pages. Minor corrections, added Appendix B on well-posedness in
classical regime. To appear in Stoch PDE: Anal Com
Rough path theory
Full text linkThe theory of rough paths arose from a desire to establish continuity
properties of ordinary differential equations involving terms of low
regularity. While essentially an analytic theory, its main motivation and
applications are in stochastic analysis, where it has given a new perspective
on It\^o calculus and a meaning to stochastic differential equations driven by
irregular paths outside the setting of semi-martingales. In this survey, we
present some of the main ideas that enter rough path theory. We discuss
complementary notions of solutions for rough differential equations and the
related notion of path signature, and give several applications and
generalisations of the theory.Comment: 25 pages, 4 figures. Invited contribution to Encyclopedia of
Mathematical Physics 2nd Editio
Rough Path Theory
Full text linkThe theory of rough paths arose from a desire to establish continuity properties of ordinary differential equations involving terms of low regularity. While essentially an analytic theory, its main motivation and applications are in stochastic analysis, where it has given a new perspective on Itô calculus and a meaning to stochastic differential equations driven by irregular paths outside the setting of semi-martingales. In this survey, we present some of the main ideas that enter rough path theory. We discuss complementary notions of solutions for rough differential equations and the related notion of path signature, and give several applications and generalisations of the theory.</p
Villain action in lattice gauge theory
Full text linkWe prove that Villain interaction applied to lattice gauge theory can be
obtained as the limit of both Wilson and Manton interactions on a larger graph
which we call the {\em carpet graph.} This is the lattice gauge theory analog
of a well-known property for spin models where Villain type interactions
are the limit of spin systems defined on a {\em cable
graph}.
Perhaps surprisingly in the setting of lattice gauge theory, our proof also
applies to non-Abelian lattice theory such as -lattice gauge theory and
its limiting Villain interaction.
In the particular case of an Abelian lattice gauge theory, this allows us to
extend the validity of Ginibre inequality to the case of the Villain
interaction.Comment: 17 pages, 2 figure
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Die Laplace-Methode für singuläre stochastische partielle Differentialgleichungen
No full textIn this thesis, we implement a Laplace method for the renormalised solution to the generalised 2D Parabolic Anderson Model (gPAM) driven by a small spatial white noise. This model is a prime example of a singular stochastic partial differential equation (PDE) that can be solved via Hairer’s theory of regularity structure, the theoretical underpinning of our work.
In this setting, we generalise classical ideas of Azencott and Ben Arous on path space as well as Aida and Inahama and Kawabi on rough path space to the space of models. The technical cornerstone of our argument is a Taylor expansion of the solution in the noise intensity parameter, a result that is of independent interest. In particular, we establish precise bounds for the terms and the remainder in the expansion and use them to estimate asymptotically irrevelant terms to arbitrary order. This leads to the first main contribution of the thesis at hand, an asymptotic expansion in powers of the noise intensity for Laplace functionals applied to the solution of gPAM. In particular, we recover the classical large deviation factor since our expansion provides much finer information on the asymptotic behaviour of said functionals than on the logarithmic scale.
We then show that the constant coefficient in the expansion can be represented in terms of traces and Carleman-Fredholm determinants of certain explicit operators, the second main result presented in this thesis. Our reasoning is based on the fact that the minimiser of the extended phase functional of gPAM has better than just Cameron-Martin regularity. This result is combined with classical analysis in abstract Wiener spaces but may prove useful in other contexts as well.
Most of the arguments in this thesis are not specific to gPAM and can be generalised to cover other singular stochastic PDEs within the regularity structures framework. However, some generalisations do pose technical challenges in which case we outline how to overcome them. Finally, we present some ideas how our results connect to other problems and how the assumptions they are based upon may be relaxed in future work.In dieser Arbeit implementieren wir eine Laplace-Methode für die renormierte Lösung des verallgemeinerten Parabolischen Anderson-Modells (gPAM) in zwei Dimensionen, welches von einem räumlichen Weißen Rauschen mit kleiner Intensität angetrieben wird. Dieses Modell ist eines der einfachsten Beispiele einer singulären stochastischen partiellen Differentialgleichung (PDG), die mit Hairers Theorie der Regularitätsstrukturen gelöst werden kann. Diese bildet die theoretische Grundlage der vorliegenden Arbeit.
Darauf aufbauend verallgemeinern wir klassische Ideen von Azencott und Ben Arous für den Pfadraum und von Aida und Inahama und Kawabi für den Raum rauer Pfade auf den Raum der Modelle. Das technische Fundament unseres Arguments ist eine Taylor-Entwicklung der Lösung in dem Parameter, welcher die Intensität des Rauschen moduliert. Dieses Resultat ist auch von unabhängigem Interesse. Dabei beweisen wir präzise Abschätzungen für die Terme und das Restglied in der Entwicklung und nutzen diese, um asymptotisch vernachlässigbare Terme beliebiger Ordnung abzuschätzen. Insgesamt leiten wir daraus den ersten großen Beitrag ab, den die vorliegende Arbeit liefert: Eine asymptotische Entwicklung für Laplace-Funktionale der Lösung von gPAM in Potenzen des Intensitätsparameters. Auf einer logarithmischen Skala ergibt sich daraus insbesondere der Faktor, welcher aus der klassischen Theorie der großen Abweichungen bekannt ist. Das liegt darin begründet, dass unsere Entwicklung viel genauere Informationen über das asymptotische Verhalten der vorher genannten Funktionale beinhaltet.
Anschließend zeigen wir, dass der konstante Koeffizient in der von uns gewonnenen Entwicklung mittels Spuren und Carleman-Fredholm-Determinanten gewisser expliziter Operatoren dargestellt werden kann. Das ist das zweite Hauptresultat der vorliegenden Arbeit. Unsere Argumentation basiert auf der Tatsache, dass der Minimierer des erweiterten Phasenfunktionals von gPAM bessere Regularität hat als jene im Cameron-Martin-Raum. Dieses Resultat kombinieren wir mit klassischer Analysis auf abstrakten Wiener-Räumen, aber es kann sich auch in anderen Kontexten als diesem als nützlich erweisen.
Der Großteil der Argumente in dieser Arbeit ist nicht nur auf gPAM anwendbar, sondern kann auch auf andere singuläre stochastische PDG verallgemeinert werden, welche man mit Regularitätsstruk- turen analysieren kann. Jedoch gibt es einige Argumente, deren Verallgemeinerung tatsächlich zu technischen Schwierigkeiten führt. In diesem Fall skizzieren wir, wie man diese überwinden kann. Zum Schluss präsentieren wir einige Ideen, in welcher Beziehung die Resultate dieser Arbeit zu anderen mathematischen Problemen stehen und wie man die Annahmen, die ihnen zugrunde liegen, in zukünftigen Arbeiten abschwächen kann.EC/H2020/683164/EU/Geometric aspects in pathwise stochastic analysis and related topics/GPSARTDFG, 410208580, GRK 2544: Stochastic Analysis in Interactio
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Mini-Workshop: Combinatorial and Algebraic Structures in Rough Analysis and Related Fields
Full text linkRecent years have seen an explosion of algebraic methods to study singular stochastic and rough dynamics. These include developments in geometric rough path theory based on the algebra of words, the introduction of decorated trees in regularity structures, and the recent approach to singular stochastic partial differential equations based on multi-indices. These developments have furthermore led to important links with numerical analysis, machine learning, stochastic quantisation, and the study of symmetries of physical systems. The aim of this mini-workshop was to bring together experts working on these fields using algebraic structures that appear in rough dynamics. The goal was to facilitate the exchange of ideas and encourage further connections to be established
Gauge field marginal of an Abelian Higgs model
Full text linkWe study the gauge field marginal of an Abelian Higgs model with Villain action defined on a 2D lattice in finite volume. Our first main result, which holds for gauge theories on arbitrary finite graphs and does not assume that the structure group is Abelian, is a loop expansion of the Radon--Nikodym derivative of the law of the gauge field marginal with respect to that of the pure gauge theory. This expansion is similar to the one of Seiler but holds in greater generality and uses a different graph theoretic approach. Furthermore, we show ultraviolet stability for the gauge field marginal of the model in a fixed gauge. More specifically, we show that moments of the H{ö}lder--Besov-type norms introduced in arXiv:1808.09196 are bounded uniformly in the lattice spacing. This latter result relies on a quantitative diamagnetic inequality that in turn follows from the loop expansion and elementary properties of Gaussian random variables.Minor fixes and additional clarification and figures added. Published versio
Invariant measure and universality of the 2D Yang-Mills Langevin dynamic
Full text linkWe prove that the Yang-Mills (YM) measure for the trivial principal bundle
over the two-dimensional torus, with any connected, compact structure group, is
invariant for the associated renormalised Langevin dynamic. Our argument relies
on a combination of regularity structures, lattice gauge-fixing, and Bourgain's
method for invariant measures. Several corollaries are presented including a
gauge-fixed decomposition of the YM measure into a Gaussian free field and an
almost Lipschitz remainder, and a proof of universality for the YM measure that
we derive from a universality for the Langevin dynamic for a wide class of
discrete approximations. The latter includes standard lattice gauge theories
associated to Wilson, Villain, and Manton actions. An important step in the
argument, which is of independent interest, is a proof of uniqueness for the
mass renormalisation of the gauge-covariant continuum Langevin dynamic, which
allows us to identify the limit of discrete approximations. This latter result
relies on Euler estimates for singular SPDEs and for Young ODEs arising from
Wilson loops.Comment: 157 pages. Shortened the earlier version. Strengthened uniqueness
result in Sec 8 which allows simplifications in Sec 3 and 5.1. Simplified Sec
5.5. Minor corrections elsewhere in the pape
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