Absolutely Continuous Research Articles

Absolute continuity of the solution to stochastic generalized Burgers–Huxley equation

Absolute continuity of the solution to stochastic generalized Burgers–Huxley equation

On the Spectrum of Certain Hadamard Manifolds

We show the absolute continuity of the spectrum and determine the spectrum as a set for two classes of Hadamard manifolds and for specific domains and quotients of one of the classes.

Remarks on absolute continuity of positive operators

Ando studied Lebesgue decomposition for (bounded) positive operators, where a notion known as the maximal absolutely continuous part plays a crucial role. Based on technique involving unbounded operators we study certain expressions for the maximal absolutely continuous part, Radon–Nikodym type results, behavior of the maximal absolutely continuous part under the tensor product operation and characterization of mutual absolute continuity and so on.

Overlapping Iterated Function Systems from the Perspective of Metric Number Theory

In this paper we develop a new approach for studying overlapping iterated function systems. This approach is inspired by a famous result due to Khintchine from Diophantine approximation which shows that for a family of limsup sets, their Lebesgue measure is determined by the convergence or divergence of naturally occurring volume sums. For many parameterised families of overlapping iterated function systems, we prove that a typical member will exhibit similar Khintchine like behaviour. Families of iterated function systems that our results apply to include those arising from Bernoulli convolutions, the{0,1,3}\{0,1,3\}problem, and affine contractions with varying translation parameter. As a by-product of our analysis we obtain new proofs of some well known results due to Solomyak on the absolute continuity of Bernoulli convolutions, and when the attractor in the{0,1,3}\{0,1,3\}problem has positive Lebesgue measure.For eacht∈[0,1]t\in [0,1]we letΦt\Phi _tbe the iterated function system given byΦt≔{ϕ1(x)=x2,ϕ2(x)=x+12,ϕ3(x)=x+t2,ϕ4(x)=x+1+t2}.\begin{equation*} \Phi _{t}≔\Big \{\phi _1(x)=\frac {x}{2},\phi _2(x)=\frac {x+1}{2},\phi _3(x)=\frac {x+t}{2},\phi _{4}(x)=\frac {x+1+t}{2}\Big \}. \end{equation*}We prove that eitherΦt\Phi _tcontains an exact overlap, or we observe Khintchine like behaviour. Our analysis shows that by studying the metric properties of limsup sets, we can distinguish between the overlapping behaviour of iterated function systems in a way that is not available to us by simply studying properties of self-similar measures.Last of all, we introduce a property of an iterated function system that we call being consistently separated with respect to a measure. We prove that this property implies that the pushforward of the measure is absolutely continuous. We include several explicit examples of consistently separated iterated function systems.

Fractional divergence-measure fields, Leibniz rule and Gauss–Green formula

AbstractGiven $$\alpha \in (0,1]$$ α ∈ ( 0 , 1 ] and $$p\in [1,+\infty ]$$ p ∈ [ 1 , + ∞ ] , we define the space $${\mathcal {D}}{\mathcal {M}}^{\alpha ,p}({\mathbb {R}}^n)$$ D M α , p ( R n ) of $$L^p$$ L p vector fields whose $$\alpha $$ α -divergence is a finite Radon measure, extending the theory of divergence-measure vector fields to the distributional fractional setting. Our main results concern the absolute continuity properties of the $$\alpha $$ α -divergence-measure with respect to the Hausdorff measure and fractional analogues of the Leibniz rule and the Gauss–Green formula. The sharpness of our results is discussed via some explicit examples.

Anisotropic ball Campanato-type function spaces and their applications

Let A be a general expansive matrix and let X be a ball quasi-Banach function space on $${\mathbb {R}}^n$$ , whose certain power (namely its convexification) supports a Fefferman–Stein vector-valued maximal inequality and the associate space of whose other power supports the boundedness of the powered Hardy–Littlewood maximal operator. The authors first introduce some anisotropic ball Campanato-type function spaces associated with both A and X, prove that these spaces are dual spaces of anisotropic Hardy spaces $$H_X^A({\mathbb {R}}^n)$$ associated with both A and X, and obtain various anisotropic Littlewood–Paley function characterizations of $$H_X^A({\mathbb {R}}^n)$$ . Also, as applications, the authors establish several equivalent characterizations of anisotropic ball Campanato-type function spaces, which, combined with the atomic decomposition of tent spaces associated with both A and X, further induce their Carleson measure characterization. All these results have a wide range of generality and, particularly, even when they are applied to Morrey spaces and Orlicz-slice spaces, some of the obtained results are also new. The novelties of this article are reflected in that, to overcome the essential difficulties caused by the absence of both an explicit expression and the absolute continuity of the quasi-norm $$\Vert \cdot \Vert _X$$ under consideration, the authors embed X under consideration into the anisotropic weighted Lebesgue space with certain special weight and then fully use the known results of this weighted Lebesgue space.

The Green function with pole at infinity applied to the study of the elliptic measure

In $\mathbb R^{d+1}_+$ or in $\mathbb R^n\setminus \mathbb R^d$ ($d<n-1$), we study the Green function with pole at infinity introduced by David, Engelstein, and Mayboroda. In two cases, we deduce the equivalence between the elliptic measure and the Lebesgue measure on $\mathbb R^d$; and we further prove the $A_\infty$-absolute continuity of the elliptic measure for operators that can be related to the two previous cases via Carleson measures, extending the range of operators for which the $A_\infty$-absolute continuity of the elliptic measure is known.

Stability of the Non-Critical Spectral Properties I: Arithmetic Absolute Continuity of the Integrated Density of States

Stability of the Non-Critical Spectral Properties I: Arithmetic Absolute Continuity of the Integrated Density of States

Quadratic Signaling With Prior Mismatch at an Encoder and Decoder: Equilibria, Continuity, and Robustness Properties

We consider communications through a Gaussian noise channel between an encoder and a decoder which have subjective probabilistic models on the source distribution. Although they consider the same cost function, the induced expected costs are misaligned due to their prior mismatch, which requires a game-theoretic approach. We consider two approaches: a Nash setup, with no prior commitment, and a Stackelberg solution concept, where the encoder is committed to a given announced policy <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">a priori</i> . We show that the Stackelberg equilibrium cost of the encoder is upper semicontinuous, under the Wasserstein metric, as encoder’s prior approaches the decoder’s prior, and it is also lower semicontinuous with Gaussian priors. For the Stackelberg setup, the optimality of affine policies for Gaussian signaling no longer holds under prior mismatch, and thus, team-theoretic optimality of linear/affine policies are not robust to perturbations. We provide conditions under which there exist informative Nash and Stackelberg equilibria with affine policies. Finally, we show existence of fully informative Nash and Stackelberg equilibria for the cheap talk problem under an absolute continuity condition.

Spatial Tightness at the Edge of Gibbsian Line Ensembles

Consider a sequence of Gibbsian line ensembles, whose lowest labeled curves (i.e., the edge) have tight one-point marginals. Then, given certain technical assumptions on the nature of the Gibbs property and underlying random walk measure, we prove that the entire spatial process of the edge is tight. We then apply this black-box theory to the log-gamma polymer Gibbsian line ensemble, which we construct. The edge of this line ensemble is the transversal free energy process for the polymer, and our theorem implies tightness with the ubiquitous KPZ class 2/3 exponent, as well as Brownian absolute continuity of all the subsequential limits. A key technical innovation which fuels our general result is the construction of a continuous grand monotone coupling of Gibbsian line ensembles with respect to their boundary data (entrance and exit values, and bounding curves). Continuous means that the Gibbs measure varies continuously with respect to varying the boundary data, grand means that all uncountably many boundary data measures are coupled to the same probability space, and monotone means that raising the values of the boundary data likewise raises the associated measure. This result applies to a general class of Gibbsian line ensembles where the underlying random walk measure is discrete time, continuous valued and log-convex, and the interaction Hamiltonian is nearest neighbor and convex.

On the computational properties of basic mathematical notions

AbstractWe investigate the computational properties of basic mathematical notions pertaining to ${\mathbb R}\rightarrow {\mathbb R}$-functions and subsets of ${\mathbb R}$, like finiteness, countability, (absolute) continuity, bounded variation, suprema and regularity. We work in higher-order computability theory based on Kleene’s S1–S9 schemes. We show that the aforementioned italicised properties give rise to two huge and robust classes of computationally equivalent operations, the latter based on well-known theorems from the mainstream mathematics literature. As part of this endeavour, we develop an equivalent $\lambda $-calculus formulation of S1–S9 that accommodates partial objects. We show that the latter are essential to our enterprise via the study of countably based and partial functionals of type $3$.

On Absolute Continuity of Non Negative Functions

This paper is committed to the study of absolute continuity of non negative functions with respect to vector measures. Almost everywhere properties are applied to establish boundedness, measurability and convergence of sequences of measurable functions. The measurability of sets with respect to vector duality functions with values in a Hilbert space is considered.

Remarks on the effective Jordan decomposition

According to the Jordan Decomposition Theorem, for any function f:[0,1]→R of bounded variation there exists a pair of nondecreasing functions f1,f2:[0,1]→R with f=f1−f2. Such a pair is called a Jordan decomposition of f. Ko (1991) and Zheng and Rettinger (2005) have analyzed computability-theoretic and complexity-theoretic aspects of this theorem. We complement their observations by three new results. One of our results says that there exists a polynomial time computable, absolutely continuous function f:[0,1]→R with polynomial modulus of absolute continuity such that, on the one hand, the so-called standard Jordan decomposition of f is computable but, on the other hand, neither the standard Jordan decomposition of f nor any other Jordan decomposition of f can be computed easily, say, in polynomial time, or in exponential time.

Distributed optimization in predefined-time for multi-agent systems over a directed network

This paper is concerned with the distributed optimization problem (DOP) with equality constraints in predefined time for multi-agent systems (MASs) over a directed network. Firstly, a new principle with respect to the predefined-time convergence is developed for absolute continuous functions, which plays a key role in the consensus analysis later. Secondly, a novel control protocol is designed to address the solution of the considered DOP with convex objective functions. Thirdly, on the basis of Lyapunov functional approach and the proposed convergence principle, the predefined-time consensus conditions are achieved. Under the designed control protocol, the consensus state can reach feasible region in finite time, and subsequently converges to an optimal solution. Finally, the correctness of the theoretical analysis and the feasibility of the designed control strategy are verified by two illustrative simulations and an application example.

Henstock-Kurzweil Integration on Metric Spaces Revisited

We fill up some gaps in the existing literature on the Henstock-Kurzweil integration on metric measure spaces. The most important one is the choice of suitable candidates for `intervals' in metric spaces for which the conclusion of Cousin's lemma holds. We also provide some characterizations of compactness and completeness in terms of Cousin's lemma, along with some alternative proofs of a few related results. Then we improve the result regarding differentiability of the primitives of Henstock-Kurzweil integrable functions on metric spaces, and a few consequences. We propose shorter proofs of measure theoretic characterizations of the HK-integral in terms of the variational measure $V_F$ as well as the $ACG^\Delta$ functions. Finally, we present alternative proofs of some generalizations of two results on Lebesgue integration. The first one bypasses Vitali covering lemma for a result on absolute continuity, while the second one is a version of the fundamental theorem of calculus in our setting.

Absolute Continuity and Singularity of Spectra for the Flows $$T_t\otimes T_{at}$$

Absolute Continuity and Singularity of Spectra for the Flows $$T_t\otimes T_{at}$$

Multivariate ρ-quantiles: A spatial approach

By substituting an Lp loss function for the L1 loss function in the optimization problem defining quantiles, one obtains Lp-quantiles that, as shown recently, dominate their classical L1-counterparts in financial risk assessment. In this work, we propose a concept of multivariate Lp-quantiles generalizing the spatial (L1-)quantiles introduced by Probal Chaudhuri (J. Amer. Statist. Assoc. 91 (1996) 862–872). Rather than restricting to power loss functions, we actually allow for a large class of convex loss functions ρ. We carefully study existence and uniqueness of the resulting ρ-quantiles, both for a general probability measure over Rd and for a spherically symmetric one. Interestingly, the results crucially depend on ρ and on the nature of the underlying probability measure. Building on an investigation of the differentiability properties of the objective function defining ρ-quantiles, we introduce a companion concept of spatial ρ-depth, that generalizes the classical spatial depth. We study extreme ρ-quantiles and show in particular that extreme Lp-quantiles behave in fundamentally different ways for p≤2 and p>2. Finally, we establish Bahadur representation results for sample ρ-quantiles and derive their asymptotic distributions. Throughout, we impose only very mild assumptions on the underlying probability measure, and in particular we never assume absolute continuity with respect to the Lebesgue measure.

Absolute Continuity of the Harmonic Measure on Low Dimensional Rectifiable Sets

In the past decades, we learnt that uniform rectifiability is often a right candidate to go past Lipschitz boundaries in boundary value problems. If Omega is an open domain in mathbb {R}^n with mild topological conditions, we can even characterize the n-1 dimensional uniformly rectifiability of the boundary partial Omega by the A_infty -absolute continuity of the harmonic measure on partial Omega with respect to the surface measure. In low dimension (d<n-1), David and Mayboroda tackled one direction of the above characterization, i.e. proved that if Gamma is a d-dimensional uniformly rectifiable set, then the harmonic measure (associated to an suitable degenerate elliptic operator) on Gamma is A_infty -absolutely continuous with respect to the d-dimensional Hausdorff measure. In the present article, we use a completely new approach to give an alternative and significantly shorter proof of David and Mayboroda’s result.

Distortion theorems for homeomorphic Sobolev mappings of integrable p-dilatations

"We study the distortion features of homeomorphisms of Sobolev class $W^{1,1}_{\rm loc}$ admitting integrability for $p$-outer dilatation. We show that such mappings belong to $W^{1,n-1}_{\rm loc},$ are differentiable almost everywhere and possess absolute continuity in measure. In addition, such mappings are both ring and lower $Q$-homeomorphisms with appropriate measurable functions $Q.$ This allows us to derive various distortion results like Lipschitz, H\""older, logarithmic H\""older continuity, etc. We also establish a weak bounded variation property for such class of homeomorphisms."

Integrable and absolutely continuous vector-valued functions

The theory of integration for functions with values in a topological vector space is a bit of a messy subject. There are many different choices for the integral, most distinct from one another in general, and with no compelling reason to always adopt one definition or another as the “right” one. No attempt is made here to resolve this question of which integral is the “right” integral, but we overview a few of the common notions and prove a couple of useful properties for the notion of integrability by seminorm. For this notion of integrability, the completeness of the space of integrable functions with values in a complete locally convex space is established. This space is then easily seen to be isomorphic to the completion of the projective tensor product of the usual L1 space of scalar functions with the vector space. Absolute continuity for this notion of integrability is also considered. Here, it is shown that the familiar properties of differentiation for absolutely continuous scalar functions holds in the vector-valued case.