1,721,011 research outputs found
On relations between a Priori bounds for measures on configuration spaces
No full textKondratiev Y, Kuna T, Kutoviy O. On relations between a Priori bounds for measures on configuration spaces. INFINITE DIMENSIONAL ANALYSIS QUANTUM PROBABILITY AND RELATED TOPICS. 2004;7(2):195-213.Some a priori bounds for measures on configuration spaces are considered. We establish relations between them and consequences for corresponding measures (such as support properties etc.). Applications to Gibbs measures are discussed
Harmonic analysis on configuration space I. General theory
No full textWe develop a combinatorial version of harmonic analysis on configuration spaces over Riemannian manifolds. Our constructions are based on the use of a lifting operator which can be considered as a kind of (combinatorial) Fourier transform in the configuration space analysis. The latter operator gives us a natural lifting of the geometry from the underlying manifold onto the configuration space. Properties of correlation measures for given states (i.e. probability measures) on configuration spaces are studied including a characterization theorem for correlation measures
Ergodicity of canonical Gibbs measures with respect to the diffeomorphism group
No full textFor general potentials we prove that every canonical Gibbs measure on configurations over a manifold X is quasi-invariant w.r.t. the group of diffeomorphisms on X. We show that this quasi-invariance property also characterizes the class of canonical Gibbs measures. From this we conclude that the extremal canonical Gibbs measures are just the ergodic ones w.r.t. the diffeomorphism group. Thus we provide a whole class of different irreducible representations. © 2004 WILEY-VCH Verlag GmbH & Co. KGaA. Weinheim
Almost sure error bounds for data assimilation in dissipative systems with unbounded observation noise
No full textData assimilation is uniquely challenging in weather forecasting due to the high dimensionality of the employed models and the nonlinearity of the governing equations. Although current operational schemes are used successfully, our understanding of their long-term error behavior is still incomplete. In this work, we study the error of some simple data assimilation schemes in the presence of unbounded (e.g., Gaussian) noise on a wide class of dissipative dynamical systems with certain properties, including the Lorenz models and the two-dimensional incompressible Navier-Stokes equations. We exploit the properties of the dynamics to derive analytic bounds on the long-term error for individual realizations of the noise in time. These bounds are proportional to the variance of the noise. Furthermore, we find that the error exhibits a form of stationary behavior, and in particular an accumulation of error does not occur. This improves on previous results in which either the noise was bounded or the error was considered in expectation only
Generalized early warning signals in multivariate and gridded data with an application to tropical cyclones
No full textTipping events in dynamical systems have been studied across many applications, often by measuring changes in variance or autocorrelation in a one-dimensional time series. In this paper, methods for detecting early warning signals of tipping events in multidimensional systems are reviewed and expanded. An analytical justification of the use of dimension-reduction by empirical orthogonal functions, in the context of early warning signals, is provided and the one-dimensional techniques are also extended to spatially separated time series over a 2D field. The challenge of predicting an approaching tropical cyclone by a tipping-point analysis of the sea-level pressure series is used as the primary example, and an analytical model of a moving cyclone is also developed in order to test predictions. We show that the one-dimensional power spectrum indicator may be used following dimension-reduction or over a 2D field. We also show the validity of our moving cyclone model with respect to tipping-point indicators
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The full moment problem on subsets of probabilities and point configurations
Full text linkThe aim of this paper is to study the full K−moment problem for measures supported on some particular infinite dimensional non-linear spaces K. We focus on the case of random measures, that is K is a subset of all non-negative Radon measures on
R
d
. We consider as K the space of sub-probabilities, probabilities and point configurations on
R
d
. For each of these spaces we provide at least one representation as a generalized basic closed semi-algebraic set to apply the main result in [J. Funct. Anal., 267 (2014) no.5: 1382–1418]. We demonstrate that this main result can be significantly improved by further considerations based on the particular chosen representation of K. In the case when K is a space of point configurations, the correlation functions (also known as factorial moment functions) are easier to handle than the ordinary moment functions. Hence, we additionally express the main results in terms of correlation functions
A moment problem for random discrete measures
No full textLet X be a locally compact Polish space. A random measure on X is a probability measure on the space of all (nonnegative) Radon measures on X. Denote by K(X) the cone of all Radon measures η on X which are of the form η=∑isiδxi, where, for each i, si>0 and δ xi is the Dirac measure at xiεX. A random discrete measure on X is a probability measure on K(X). The main result of the paper states a necessary and sufficient condition (conditional upon a mild a priori bound) when a random measure μ is also a random discrete measure. This condition is formulated solely in terms of moments of the random measure μ. Classical examples of random discrete measures are completely random measures and additive subordinators, however, the main result holds independently of any independence property. As a corollary, a characterization via moments is given when a random measure is a point process
Linear and fractional response for the SRB measure of smooth hyperbolic attractors and discontinuous observables
No full textWe consider a smooth one-parameter family t → ( ft : M → M) of diffeomorphisms with compact transitive Axiom A attractors λt denoting by dpt the SRB measure of fttλt. Our first result is that for any function θ in the Sobolev space Hrp(M), with 1π-rfpagπ ∞ and 0 π r π 1/p, the map tx→ ∫ θ dpt is ?-Hölder continuous for all r. This applies to(x) = h(x) θ (g(x) ? a) (for all >1) for h and g smooth and θ the Heaviside function, if a is not a critical value of g. Our second result says that for any such function -(x) = h(x) θ (g(x) ? a) so that in addition the intersection of {x|g(x) = a} with the support of h is foliated by admissible stable leaves of ft, the map t d-t is differentiable. (We provide distributional linear response and fluctuation-dissipation formulas for the derivative.) Obtaining linear response or fractional response for such observables θ is motivated by extreme-value theory
Power spectrum scaling as a measure of critical slowing down and precursor to tipping points in dynamical systems
Full text linkMany dynamical systems experience sudden shifts in behaviour known as tipping points or critical transitions, often preceded by the 'critical slowing down' (CSD) phenomenon whereby the recovery times of a system increase as the tipping point is approached. Many attempts have been made to find a tipping point indicator: a proxy for CSD, such that a change in the indicator acts as an early warning signal. Several generic tipping point indicators have been suggested, these include the power spectrum (PS) scaling exponent whose use as an indicator has previously been justified by its relationship to the well-established detrended fluctuation analysis (DFA) exponent. In this paper we justify the use of the PS indicator analytically, by considering a mathematical formulation of the CSD phenomenon. We assess the usefulness of estimating the PS scaling exponent in a tipping point context when the PS does not exhibit power-law scaling, or changes over time. In addition we show that this method is robust against trends and oscillations in the time series, making it a good candidate for studying resilience of systems with periodic oscillations which are observed in ecology and geophysics
The full infinite dimensional moment problem on semi-algebraic sets of generalized functions
No full textWe consider a generic basic semi-algebraic subset S of the space of generalized functions, that is a set given by (not necessarily countably many) polynomial constraints. We derive necessary and sufficient conditions for an infinite sequence of generalized functions to be realizable on S, namely to be the moment sequence of a finite measure concentrated on S. Our approach combines the classical results about the moment problem on nuclear spaces with the techniques recently developed to treat the moment problem on basic semi-algebraic sets of Rd. In this way, we determine realizability conditions that can be more easily verified than the well-known Haviland type conditions. Our result completely characterizes the support of the realizing measure in terms of its moments. As concrete examples of semi-algebraic sets of generalized functions, we consider the set of all Radon measures and the set of all the measures having bounded Radon-Nikodym density w.r.t. the Lebesgue measure. © 2014 Elsevier Inc
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