Measure Space Research Articles

Viscosity solutions of centralized control problems in measure spaces

This work focuses on a control problem in the Wasserstein space of probability measures over $\mathbb{R}^d$. Our aim is to link this control problem to a suitable Hamilton-Jacobi-Bellman (HJB) equation. We explore a notion of viscosity solution using test functions that are locally Lipschitz and locally semiconvex or semiconcave functions. This regularity allows to define a notion of viscosity and a Hamiltonian function relying on directional derivatives. Using a generalization of Ekeland's principle, we show that the corresponding HJB equation admits a comparison principle, and deduce that the value function is the unique solution in this viscosity sense. The PDE tools are developed in the general framework of Measure Differential Equations.

Compactification of spaces of measures and pseudocompactness

We prove pseudocompactness of a Tychonoff space X and the space P(X) of Radon probability measures on it with the weak topology under the condition that the Stone–ech compactification of the space P(X) is homeomorphic to the space P(βX) of Radon probability measures on the Stone–ech compactification of the space X.

Towards a Common Metrics and Evaluation Framework for Assessment of Older Adults and Caregivers Interacting with Artificial Intelligence

Artificial intelligence (AI) has applications in assisting older adults to age in place and provide support to them and their caregivers as their cognition declines with age. However, effective assessment methods of this technology are needed in order to benchmark their performance and a common set of metrics and evaluation methods would enable such assessments to be compared to one another. To this end, we propose a common framework for human-AI interaction involving care recipients and their care networks. From the results of a literature review exercise, a framework with sample metrics, related measures, qualified evaluation tools, and contextual factors that impact assessment are reviewed. This paper provides a sample of common metrics in one of the framework’s measurement spaces (human-AI interaction) and discusses some of the impacts of contextual factors and how use of the common metrics and evaluation framework can be used for meta-analysis and to guide future research. Additional future articles are planned to cover the other measurement spaces in the framework (system performance, task performance, and well-being), including their particular common metrics and evaluation methods. This effort aims to provide guidance for researchers in this domain as well as highlight measurement gaps that can be filled by future research.

SVI solutions to stochastic nonlinear diffusion equations on general measure spaces

SVI solutions to stochastic nonlinear diffusion equations on general measure spaces

Application of Very Low Frequency (VLF) Method for Underground River Estimation in Donorojo Sub-District, Pacitan

<p class="Keywords"><span lang="EN-US">Pacitan is included in the Gunung Sewu karst area in the Southern Mountain Zone, and carbonate rocks dominate the constituent rocks. Karst has a unique drainage system because it is dominated by subsurface flow. This research was conducted in Cemeng and Klepu villages, Donorojo sub-district, Pacitan, with 4 track data. Data were collected using the very low frequency electromagnetic (VLF-EM) method with a track length ranging from 200-450 m, a measurement spacing of 5 m, and a transmitter frequency of 19.8 kHz. Data processing uses filtering and inversion, resulting in a cross-section of resistivity values. Based on the subsurface resistivity cross-section profile, the cavity in the carbonate rock layer with a resistivity value of 0-500 Ωm is identified as an underground river. The underground river is found on tracks 1, 2, 3, and 4 near the surface, with a depth of about 30 m below the surface.</span></p>

Transfunctions Applied to Plans, Markov Operators and Optimal Transport

Abstract A transfunction is a function which maps between sets of finite measures on measurable spaces. In this paper we characterize transfunctions that correspond to Markov operators and to plans; such a transfunction will contain the “instructions” common to several Markov operators and plans. We also define the adjoint of transfunctions in two settings and provide conditions for existence of adjoints. Finally, we develop approximations of identity in each setting and use them to approximate weakly-continuous transfunctions with simple transfunctions; one of these results can be applied to some optimal transport problems to approximate the optimal cost with simple Markov transfunctions.

Metric measure spaces with Ricci bounds from below

In this survey we introduce new results on metric measure spaces with Ricci curvature bounded below, so-called R C D \mathrm {RCD} spaces, including stability results of several energies with respect to measured Gromov-Hausdorff convergence, Weyl’s law, and a characterization of noncollapsed RCD spaces.

Shadowing of the induced map for contracting homeomorphisms

Let f:X→X be a contracting homeomorphism of a metric space with positive diameter. We prove that the induced map f⁎ in the space of probability measures equipped with the Prokhorov metric does not have the shadowing property. However, if X is Polish, then the restriction of f⁎ to the Wasserstein space has the generalized shadowing property as per Boyarsky and Gora [4], concerning the Kantorovich-Rubinstein and Prokhorov metrics.

Metric compatibility and determination in complete metric spaces

AbstractThe local slope operator was introduced in De Giorgi et al. (Atti Accad Naz Lincei Rend Cl Sci Fis Mat Nat 68:180–187, 1980) to study gradient flow dynamics in metric spaces. This tool has now become a cornerstone in metric evolution equations (see, e.g., Ambrosio et al. (Gradient flows in metric spaces and in the space of probability measures, Lectures in Mathematics. Birkhauser, 2008). Very recently, in Daniilidis and Salas (Proc Am Math Soc 150:4325–4333, 2022), it was established that Lipschitz inf-compact functions are uniquely determined by their local slope and critical values. Compactness played a paramount role in this result, ensuring in particular the existence of critical points. We hereby emancipate from this restriction and establish a determination result for merely bounded from below functions, by adding an assumption controlling the asymptotic behavior. This assumption is trivially fulfilled if f is inf-compact. In addition, our result is not only valid for the (De Giorgi) local slope, but also for the main paradigms of average descent operators as well as for the global slope, case in which the asymptotic assumption becomes superfluous. Therefore, the present work extends simultaneously the metric determination results of Daniilidis and Salas (Proc Am Math Soc 150:4325–4333, 2022) and Thibault and Zagrodny (Commun Contemp Math 25(7):2250014, 2023).

Kolmogorov equations on the space of probability measures associated to the nonlinear filtering equation: the viscosity approach

. We study the backward Kolmogorov equation on the space of probability measures associated to the Kushner-Stratonovich equation of nonlinear filtering. We prove existence and uniqueness in the viscosity sense and, in particular, we provide a comparison theorem. In the context of stochastic filtering, it is natural to consider measure-valued processes that satisfy stochastic differential equations. In the literature, a classical way to address this problem is by assuming that these measure-valued processes admit a density. Our approach is different and we work directly with measures. Thus, the backward Kolmogorov equation we study is a second-order partial differential equation of parabolic type on the space of probability measures with compact support. In the literature, only few results are available on viscosity solutions for this kind of problems and in particular the uniqueness is a very challenging issue. Here we find a viscosity solution and then we prove that it is unique via comparison theorem.

Some analytic and geometric aspects of weighted p-Laplacian on smooth metric measure spaces

Some analytic and geometric aspects of weighted <i>p</i>-Laplacian on smooth metric measure spaces

Measurable Krylov spaces and eigenenergy count in quantum state dynamics

In this work, we propose a quantum-mechanically measurable basis for the computation of spread complexity. Current literature focuses on computing different powers of the Hamiltonian to construct a basis for the Krylov state space and the computation of the spread complexity. We show, through a series of proofs, that time-evolved states with different evolution times can be used to construct an equivalent space to the Krylov state space used in the computation of the spread complexity. Afterwards, we introduce the effective dimension, which is upper-bounded by the number of pairwise distinct eigenvalues of the Hamiltonian. The computation of the spread complexity requires knowledge of the Hamiltonian and a classical computation of the different powers of the Hamiltonian. The computation of large powers of the Hamiltonian becomes increasingly difficult for large systems. The first part of our work addresses these issues by defining an equivalent space, where the original basis consists of quantum-mechanically measurable states. We demonstrate that a set of different time-evolved states can be used to construct a basis. We subsequently verify the results through numerical analysis, demonstrating that every time-evolved state can be reconstructed using the defined vector space. Based on this new space, we define an upper-bounded effective dimension and analyze its influence on finite-dimensional systems. We further show that the Krylov space dimension is equal to the number of pairwise distinct eigenvalues of the Hamiltonian, enabling a method to determine the number of eigenenergies the system has experimentally. Lastly, we compute the spread complexities of both basis representations and observe almost identical behavior, thus enabling the computation of spread complexities through measurements.

Li-Yau type estimation of a semilinear parabolic system along geometric flow

This article provides a Li–Yau-type gradient estimate for a semilinear weighted parabolic system of semilinear equations along an abstract geometric flow on a smooth measure space. A Harnack-type inequality on the system is also derived at the end.

Quasi-Compactness of Operators for General Markov Chains and Finitely Additive Measures

We study Markov operators T, A, and T* of general Markov chains on an arbitrary measurable space. The operator, T, is defined on the Banach space of all bounded measurable functions. The operator A is defined on the Banach space of all bounded countably additive measures. We construct an operator T*, topologically conjugate to the operator T, acting in the space of all bounded finitely additive measures. We prove the main result of the paper that, in general, a Markov operator T* is quasi-compact if and only if T is quasi-compact. It is proved that the conjugate operator T* is quasi-compact if and only if the Doeblin condition (D) is satisfied. It is shown that the quasi-compactness conditions for all three Markov operators T, A, and T* are equivalent to each other. In addition, we obtain that, for an operator T* to be quasi-compact, it is necessary and sufficient that it does not have invariant purely finitely additive measures. A strong uniform reversible ergodic theorem is proved for the quasi-compact Markov operator T* in the space of finitely additive measures. We give all the proofs for the most general case. A detailed analysis of Lin’s counterexample is provided.

Pre-measure spaces and pre-integration spaces in predicative Bishop-Cheng measure theory

Bishop's measure theory (BMT) is an abstraction of the measure theory of a locally compact metric space $X$, and the use of an informal notion of a set-indexed family of complemented subsets is crucial to its predicative character. The more general Bishop-Cheng measure theory (BCMT) is a constructive version of the classical Daniell approach to measure and integration, and highly impredicative, as many of its fundamental notions, such as the integration space of $p$-integrable functions $L^p$, rely on quantification over proper classes (from the constructive point of view). In this paper we introduce the notions of a pre-measure and pre-integration space, a predicative variation of the Bishop-Cheng notion of a measure space and of an integration space, respectively. Working within Bishop Set Theory (BST), and using the theory of set-indexed families of complemented subsets and set-indexed families of real-valued partial functions within BST, we apply the implicit, predicative spirit of BMT to BCMT. As a first example, we present the pre-measure space of complemented detachable subsets of a set $X$ with the Dirac-measure, concentrated at a single point. Furthermore, we translate in our predicative framework the non-trivial, Bishop-Cheng construction of an integration space from a given measure space, showing that a pre-measure space induces the pre-integration space of simple functions associated to it. Finally, a predicative construction of the canonically integrable functions $L^1$, as the completion of an integration space, is included.

Compactification of Spaces of Measures and Pseudocompactness

Compactification of Spaces of Measures and Pseudocompactness

Relativistic imprints on dispersion measure space distortions

Relativistic imprints on dispersion measure space distortions

Further remarks on absorbing Markov decision processes

In this note, based on the recent remarkable results of Dufour and Prieto-Rumeau, we deduce that for an absorbing Markov decision process with a given initial state, under a standard compactness-continuity condition, the space of occupation measures has the same convergent sequences, when it is endowed with the weak topology and with the weak-strong topology. We provided two examples demonstrating that imposed condition cannot be replaced with its popular alternative, and the above assertion does not hold for the space of marginals of occupation measures on the state space. Moreover, the examples also clarify some results in the previous literature.

$$A_p$$ weights on nonhomogeneous trees equipped with measures of exponential growth

AbstractThis paper aims to study $$A_p$$ A p weights in the context of a class of metric measure spaces with exponential volume growth, namely infinite trees with root at infinity equipped with the geodesic distance and flow measures. Our main result is a Muckenhoupt Theorem, which is a characterization of the weights for which a suitable Hardy–Littlewood maximal operator is bounded on the corresponding weighted $$L^p$$ L p spaces. We emphasise that this result does not require any geometric assumption on the tree or any condition on the flow measure. We also prove a reverse Hölder inequality in the case when the flow measure is locally doubling. We finally show that the logarithm of an $$A_p$$ A p weight is in BMO and discuss the connection between $$A_p$$ A p weights and quasisymmetric mappings.

Wasserstein-Kaplan-Meier Survival Regression

Survival analysis plays a pivotal role in medical research, offering valuable insights into the timing of events such as survival time. One common challenge in survival analysis is the necessity to adjust the survival function to account for additional factors, such as age, gender, and ethnicity. We propose an innovative regression model for right-censored survival data across heterogeneous populations, leveraging the Wasserstein space of probability measures. Our approach models the probability measure of survival time and the corresponding non-parametric Kaplan-Meier estimator for each subgroup as elements of the Wasserstein space. The Wasserstein space provides a flexible framework for modeling heterogeneous populations, allowing us to capture complex relationships between covariates and survival times. We address an underexplored aspect by deriving the non-asymptotic convergence rate of the Kaplan-Meier estimator to the underlying probability measure in terms of the Wasserstein metric. The proposed model is supported with a solid theoretical foundation including pointwise and uniform convergence rates, along with an efficient algorithm for model fitting. The proposed model effectively accommodates random variation that may exist in the probability measures across different subgroups, demonstrating superior performance in both simulations and two case studies compared to the Cox proportional hazards model and other alternative models.