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This book gives a comprehensive and self-contained introduction to the theory of symmetric Markov processes and symmetric quasi-regular Dirichlet forms. In a detailed and accessible manner, Zhen-Qing Chen and Masatoshi Fukushima cover the essential elements and applications of the theory of symmetric Markov processes, including recurrence/transience criteria, probabilistic potential theory, additive functional theory, and time change theory. The authors develop the theory in a general framework of symmetric quasi-regular Dirichlet forms in a unified manner with that of regular Dirichlet forms, emphasizing the role of extended Dirichlet spaces and the rich interplay between the probabilistic and analytic aspects of the theory. Chen and Fukushima then address the latest advances in the theory, presented here for the first time in any book. Topics include the characterization of time-changed Markov processes in terms of Douglas integrals and a systematic account of reflected Dirichlet spaces, and the important roles such advances play in the boundary theory of symmetric Markov processes. This volume is an ideal resource for researchers and practitioners, and can also serve as a textbook for advanced graduate students. It includes examples, appendixes, and exercises with solutions.

This paper is concerned with an optimization problem which is governed by the Kantorovich problem of optimal transport. More precisely, we consider a bilevel optimization problem with the underlying problem being the Kantorovich problem. This task can be reformulated as a mathematical problem with complementarity constraints in the space of regular Borel measures. Because of the non-smoothness that is induced by the complementarity constraints, problems of this type are often regularized, e.g., by an entropic regularization. However, in this paper we apply a quadratic regularization to the Kantorovich problem. By doing so, we are able to drastically reduce its dimension while preserving the sparsity structure of the optimal transportation plan as much as possible. As the title indicates, this is the second part in a series of three papers. While the existence of optimal solutions to both the bilevel Kantorovich problem and its regularized counterpart were shown in the first part, this paper deals with the (weak-∗) convergence of solutions to the regularized bilevel problem to solutions of the original bilevel Kantorovich problem for vanishing regularization parameters.

The main focus of this work is on providing a formal definition of statistical depth for functional data on the basis of six properties, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth defining properties is one that addresses the delicate challenge of inherent partial observability of functional data, with fulfillment giving rise to a minimal guarantee on the performance of the empirical depth beyond the idealised and practically infeasible case of full observability. As an incidental product, functional depths satisfying our definition achieve a robustness that is commonly ascribed to depth, despite the absence of a formal guarantee in the multivariate definition of depth. We demonstrate the fulfillment or otherwise of our properties for six widely used functional depth proposals, thereby providing a systematic basis for selection of a depth function.

Let $\\mu$ be a Radon measure on the nth Heisenberg group ${\\mathbb{H}}^n$ . In this note we prove that if the $(2n+1)$ -dimensional (Heisenberg) Riesz transform on ${\\mathbb{H}}^n$ is $L^2(\\mu)$ -bounded, and if $\\mu(F)=0$ for all Borel sets with ${\\text{dim}}_H(F)\\leq 2$ , then $\\mu$ must have $(2n+1)$ -polynomial growth. This is the Heisenberg counterpart of a result of Guy David from [ Dav91 ].

In this paper, we follow and extend a group-theoretic method introduced by Greenleaf–Iosevich–Liu–Palsson (GILP) to study finite points configurations spanned by Borel sets in $\\mathbb{R}^n,n\\geq 2,n\\in\\mathbb{N}.$ We remove a technical continuity condition in a GILP’s theorem in [Revista Mat. Iberoamer 31 (2015), 799–810]. This allows us to extend the Wolff–Erdogan dimension bound for distance sets to finite points configurations with k points for $k\\in\\{2,\\dots,n+1\\}$ forming a $(k-1)$ -simplex.

A set $U \\subseteq {\\mathbb {R}} \\times {\\mathbb {R}}$ is universal for countable subsets of ${\\mathbb {R}}$ if and only if for all $x \\in {\\mathbb {R}}$ , the section $U_x = \\{y \\in {\\mathbb {R}} : U(x,y)\\}$ is countable and for all countable sets $A \\subseteq {\\mathbb {R}}$ , there is an $x \\in {\\mathbb {R}}$ so that $U_x = A$ . Define the equivalence relation $E_U$ on ${\\mathbb {R}}$ by $x_0 \\ E_U \\ x_1$ if and only if $U_{x_0} = U_{x_1}$ , which is the equivalence of codes for countable sets of reals according to U. The Friedman–Stanley jump, $=^+$ , of the equality relation takes the form $E_{U^*}$ where $U^*$ is the most natural Borel set that is universal for countable sets. The main result is that $=^+$ and $E_U$ for any U that is Borel and universal for countable sets are equivalent up to Borel bireducibility. For all U that are Borel and universal for countable sets, $E_U$ is Borel bireducible to $=^+$ . If one assumes a particular instance of $\\mathbf {\\Sigma }_3^1$ -generic absoluteness, then for all $U \\subseteq {\\mathbb {R}} \\times {\\mathbb {R}}$ that are $\\mathbf {\\Sigma }_1^1$ (continuous images of Borel sets) and universal for countable sets, there is a Borel reduction of $=^+$ into $E_U$ .

In this paper, we present a generalized form of the Cantor ternary set by studying the cantor    middle set where 1    and its fractal dimension. The paper also introduces the Heine-Borel set and shows that the cantor set and its generalised   middle set where 1   are Heine-Borel sets.

We show that the uniform measure-theoretic ergodic decomposition of a countable Borel equivalence relation $(X, E)$ may be realized as the topological ergodic decomposition of a continuous action of a countable group $\\Gamma \\curvearrowright X$ generating E. We then apply this to the study of the cardinal algebra $\\mathcal {K}(E)$ of equidecomposition types of Borel sets with respect to a compressible countable Borel equivalence relation $(X, E)$ . We also make some general observations regarding quotient topologies on topological ergodic decompositions, with an application to weak equivalence of measure-preserving actions.

Let $\\bf\\Gamma$ be a Borel class, or a Wadge class of Borel sets, and $2\\!\\leq\\! d\\!\\leq\\!\\omega$ be a cardinal. A Borel subset $B$ of ${\\mathbb R}^d$ is potentially in $\\bf\\Gamma$ if there is a finer Polish topology on $\\mathbb R$ such that $B$ is in $\\bf\\Gamma$ when ${\\mathbb R}^d$ is equipped with the new product topology. The author provides a way to recognize the sets potentially in $\\bf\\Gamma$ and applies this to the classes of graphs (oriented or not), quasi-orders and partial orders.

For a positive finite Borel measure μ compactly supported in the complex plane, the space P2(μ) is the closure of the analytic polynomials in the Lebesgue space L2(μ). According to Thomson’s famous result, any space P2(μ) decomposes as an orthogonal sum of pieces which are essentially analytic, and a residual L2-space. We study the structure of this decomposition for a class of Borel measures μ supported on the closed unit disk D¯ for which the part μⅅ, living in the open disk ⅅ, is radial and decreases at least exponentially fast near the boundary of the disk. For the considered class of measures, we give a precise form of the Thomson decompsition. In particular, we confirm a conjecture of Kriete and MacCluer from 1990, which gives an analog to Szegö’s classical theorem.