arXiv subjects
Publications and source records attributed to Tobias Kuna.
Given a classical gas described by the truncated correlation functions of all orders, we prove convergence of an expansion of the pair interaction part of the (unknown) potential in terms of the truncated correlation functions of all orders, at infinite volume.
We investigate a coupled atmosphere-ocean model including the mechanical and thermodynamical interaction between the two fluids for the mid-latitudes. The formulation combines a multilayer quasi-geostrophic dynamical framework with temperature equations incorporating long- and short-wave radiative forcing, as in energy balance models. Within a suitable functional framework, we establish the existence and uniqueness of solutions, and their continuous dependence on the radiation parameters. We also prove that the long-time dynamics are described by a finite-dimensional global attractor and, moreover, that the system possesses a finite set of determining modes that governs its asymptotic behaviour. In particular, we show that the long-term evolution of the ocean's temperature can be reconstructed solely from observations of the velocity fields across the model's layers.
This article considers the problem of reconstructing unknown driving forces based on incomplete knowledge of the system and its state. This is studied in both a linear and nonlinear setting that is paradigmatic in geophysical fluid dynamics and various applications. Two algorithms are proposed to address this problem: one that iteratively reconstructs forcing and another that provides a continuous-time reconstruction. Convergence is shown to be guaranteed provided that observational resolution is sufficiently high and algorithmic parameters are properly tuned according to the prior information; these conditions are quantified precisely. The class of reconstructable forces identified here include those which are time-dependent and potentially inject energy at all length scales. This significantly expands upon the class of forces in previous studies, which could only accommodate those with band-limited spectra. The second algorithm moreover provides a conceptually streamlined approach that allows for a more straightforward analysis and simplified practical implementation.
An infinite particle system of independent jumping particles in infinite volume is considered. Their construction is recalled,further properties are derived, the relation with hierarchical equations, Poissonian analysis, and second quantization are discussed. The hydrodynamic limit for a general initial distribution satisfying a mixing condition is derived. The long time asymptotic is computed under an extra assumption. The relation with constructions based on infinite volume limits is discussed.
We establish a criterion for the existence of a representing Radon measure for linear functionals defined on a unital commutative real algebra $A$, which we assume to be generated by a vector space $V$ endowed with a Hilbertian seminorm $q$. Such a general criterion provides representing measures with support contained in the space of characters of $A$ whose restrictions to $V$ are $q-$continuous. This allows us in turn to prove existence results for the case when $V$ is endowed with a nuclear topology. In particular, we apply our findings to the symmetric tensor algebra of a nuclear space.
A framework to establish response theory for a class of nonlinear stochastic partial differential equations (SPDEs) is provided. More specifically, it is shown that for a certain class of observables, the averages of those observables against the stationary measure of the SPDE are differentiable (linear response) or, under weaker conditions, locally Hölder continuous (fractional response) as functions of a deterministic additive forcing. The method allows to consider observables that are not necessarily differentiable. For such observables, spectral gap results for the Markov semigroup associated with the SPDE have recently been established that are fairly accessible. This is important here as spectral gaps are a major ingredient for establishing linear response. The results are applied to the 2D stochastic Navier-Stokes equation and the stochastic two-layer quasi-geostrophic model, an intermediate complexity model popular in the geosciences to study atmosphere and ocean dynamics. The physical motivation for studying the response to perturbations in the forcings for models in geophysical fluid dynamics comes from climate change and relate to the question as to whether statistical properties of the dynamics derived under current conditions will be valid under different forcing scenarios.
We consider the class of all linear functionals $L$ on a unital commutative real algebra $A$ that can be represented as an integral w.r.t. to a Radon measure with compact support in the character space of $A$. Exploiting a recent generalization of the classical Nussbaum theorem, we establish a new characterization of this class of moment functionals solely in terms of a growth condition intrinsic to the given linear functional. To the best of our knowledge, our result is the first to exactly identify the compact support of the representing Radon measure. We also describe the compact support in terms of the largest Archimedean quadratic module on which $L$ is non-negative and in terms of the smallest submultiplicative seminorm w.r.t. which $L$ is continuous. Moreover, we derive a formula for computing the measure of each singleton in the compact support, which in turn gives a necessary and sufficient condition for the support to be a finite set. Finally, some aspects related to our growth condition for topological algebras are also investigated.
We deal with the following general version of the classical moment problem: when can a linear functional on a unital commutative real algebra $A$ be represented as an integral with respect to a Radon measure on the character space $X(A)$ of $A$ equipped with the Borel $σ-$algebra generated by the weak topology? We approach this problem by constructing $X(A)$ as a projective limit of the character spaces of all finitely generated unital subalgebras of $A$. Using some fundamental results for measures on projective limits of measurable spaces, we determine a criterion for the existence of an integral representation of a linear functional on $A$ with respect to a measure on the cylinder $σ-$algebra on $X(A)$ (resp. a Radon measure on the Borel $σ-$algebra on $X(A)$) provided that for any finitely generated unital subalgebra of $A$ the corresponding moment problem is solvable. We also investigate how to localize the support of representing measures for linear functionals on $A$. These results allow us to establish infinite dimensional analogues of the classical Riesz-Haviland and Nussbaum theorems as well as a representation theorem for linear functionals non-negative on a "partially Archimedean" quadratic module of $A$. Our results in particular apply to the case when $A$ is the algebra of polynomials in infinitely many variables or the symmetric tensor algebra of a real infinite dimensional vector space, providing a unified setting which enables comparisons between some recent results for these instances of the moment problem.
Ergodic properties of a stochastic medium complexity model for atmosphere and ocean dynamics are analysed. More specifically, a two-layer quasi-geostrophic model for geophysical flows is studied, with the upper layer being perturbed by additive noise. This model is popular in the geosciences, for instance to study the effects of a stochastic wind forcing on the ocean. A rigorous mathematical analysis however meets with the challenge that in the model under study, the noise configuration is spatially degenerate as the stochastic forcing acts only on the top layer. Exponential convergence of solutions laws to the invariant measure is established, implying a spectral gap of the associated Markov semigroup on a space of Hölder continuous functions. The approach provides a general framework for generalised coupling techniques suitable for applications to dissipative SPDEs. In case of the two-layer quasi-geostrophic model, the results require the second layer to obey a certain passivity condition.
The problem of stability of the optimal filter is revisited. The optimal filter (or filtering process) is the conditional probability of the current state of some stochastic process (the signal process), given both present and past values of another process (the observation process). Typically the filtering process satisfies a dynamical equation, and the question investigated here concerns the stability of this dynamics. In contrast to previous work, signal processes given by the iterations of a deterministic mapping $f$ are considered, with only the initial condition being random. While the stability of the filter may emerge from strong randomness of the signal processes, different and more dynamical effects will be exploited in the present work. More specifically, we consider uniformly hyperbolic $f$ with strong instabilities providing the necessary mixing. This however requires that the filtering process is initialised with densities exhibiting already a certain level of smoothness. Furthermore, $f$ may also have stable directions along which the filtering process will eventually not have a density, a major technical difficulty. Further results show that the filtering process is asymptotically concentrated on the attractor and furthermore will have densities with respect to the invariant (SRB)~measure along instable manifolds of $f$.
We prove a multivariate Lagrange-Good formula for functionals of uncountably many variables and investigate its relation with inversion formulas using trees. We clarify the cancellations that take place between the two aforementioned formulas and draw connections with similar approaches in a range of applications.
The aim of this paper is to study the full $K-$moment problem for measures supported on some particular non-linear subsets $K$ of an infinite dimensional vector space. We focus on the case of random measures, that is $K$ is a subset of all non-negative Radon measures on $\mathbb{R}^d$. We consider as $K$ the space of sub-probabilities, probabilities and point configurations on $\mathbb{R}^d$. 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.
We prove a novel inversion theorem for functionals given as power series in infinite-dimensional spaces and apply it to the inversion of the density-activity relation for inhomogeneous systems. This provides a rigorous framework to prove convergence for density functionals for inhomogeneous systems with applications in classical density function theory, liquid crystals, molecules with various shapes or other internal degrees of freedom. The key technical tool is the representation of the inverse via a fixed point equation and a combinatorial identity for trees, which allows us to obtain convergence estimates in situations where Banach inversion fails. Moreover, the new method for the inversion gives for the (homogeneous) hard sphere gas a significantly improved radius of convergence for the virial expansion improving the first and up to now best result by Lebowitz and Penrose (1964).
The semi-geostrophic equations are used widely in the modelling of large-scale atmospheric flows. In this note, we prove the global existence of weak solutions of the incompressible semi-geostrophic equations, in geostrophic coordinates, in a three-dimensional domain with a free upper boundary. The proof, based on an energy minimisation argument originally inspired by Cullen's Stability Principle, uses optimal transport results as well as the analysis of Hamiltonian ODEs in spaces of probability measures as studied by Ambrosio and Gangbo. We also give a general formulation of Cullen's Stability Principle in a rigorous mathematical framework.
Data 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 behaviour 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 2D incompressible Navier-Stokes equations. We exploit the properties of the dynamics to derive analytic bounds on the long-term error for individual realisations of the noise in time. These bounds are proportional to the amplitude of the noise. Furthermore, we find that the error exhibits a form of stationary behaviour, 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.
We consider a smooth one-parameter family $t \to f_t$ of diffeomorphisms with compact transitive Axiom A attractors. Our first result (corrected) is that for any function $G$ in the Sobolev space $H^r_p$, with $p>1$ and $0<r<1/p$, the map $R(t)$ sending $t$ to the average of $G$ with respect to the SRB measure of $f_t$ is $α$-Hölder continuous for all $α<r- |log \mathcal J|/(p|log ν_s|$) where $\mathcal J\le 1$ is the strongest volume contraction and $ν_s<1$ is the weakest contraction. This applies to $θ(x)=h(x)Θ(g(x)-a)$ (for all $α<1- |log \mathcal J|/|log ν_s|$) 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 so that, in addition, the intersection of the set of points $x$ so that $g(x)=a$ with the support of $h$ is foliated by "admissible stable leaves" of $f_t$, the map $R(t)$ is differentiable. (We provide distributional linear response and fluctuation-dissipation formulas for the derivative.) Obtaining linear response or fractional response for such observables is motivated by extreme-value theory. --- Second version, following the referee's comments: We explain better the cone choices around (2.4). Appendix A contains information on the Banach spaces. We added the paragraph containing (2.6) in the proof of Theorem 2.1. In the proof of Theorem 3.3, we do not need to introduce mollifiers. However, the new argument around (2.6) is not available here, so we must replace the pair $(u-1, |s-1|)$ by $(u-2, |s-2|)$. This is why we now assume that $h$ is $C^3$ and that $g$ and the foliations are $C^4$. --- Third version: We have added a corrigendum modifying the first result (Theorem 2.1).
We prove convergence of the multi-body correlation function as a power series in the density. We work in the context of the cluster expansion in the canonical ensemble and we obtain bounds uniform in the volume and the number of particles. In the thermodynamic limit, the coefficients are characterized by sums over some class of two-connected graphs. We introduce the "direct correlation function" in the canonical ensemble and we prove that in the thermodynamic limit it is given by a convergent power series in the density with coefficients given by sums over some other class of two-connected graphs. Furthermore, it satisfies the Ornstein-Zernike equation from which quantified approximations can be derived.
We consider a particular instance of the truncated realizability problem on the $d-$dimensional lattice. Namely, given two functions $ρ_1({\bf i})$ and $ρ_2({\bf i},{\bf j})$ non-negative and symmetric on $\mathbb{Z}^d$, we ask whether they are the first two correlation functions of a translation invariant point process. We provide an explicit construction of such a realizing process for any $d\geq 2$ when the radial distribution has a specific form. We also derive from this construction a lower bound for the maximal realizable density and compare it with the already known lower bounds.