arXiv subjects

Kaneharu Tsuchida

Publications and source records attributed to Kaneharu Tsuchida.

7 recordsLinked to original sources

arXiv2026

Energy integrals and asymmetric co-potentials for closed forms

We investigate the class of measures of finite energy integrals and the behavior of potentials and co-potentials associated with non-symmetric closed forms. In particular, we compare these objects with their symmetric counterparts from three viewpoints: a non-symmetric version of Stollmann--Voigt's inequality, non-symmetric perturbations of symmetric forms, and closed forms associated with non-symmetric jump-type forms. Our results indicate that measures of finite energy integrals, potentials, and co-potentials behave differently in the non-symmetric setting, requiring more delicate analysis than in the symmetric case.

math.PR↗

arXiv2026

Critical spectral behavior and large deviations for geometric $\alpha$-stable processes

In this paper, we study the Schr\"odinger-type operator associated with geometric stable processes on $\mathbb{R}^{d}$, especially the differentiability of spectral function. Let $\mathcal{H}$ be the generator of the geometric stable process and $\mu$ a smooth measure on $\mathbb{R}^{d}$. Then the spectral function $C(\theta)$ is defined as $C(\theta) = -\inf \sigma(-\mathcal{H} - \theta \mu)$, where $\sigma(\mathcal{A})$ denotes the spectrum of $\mathcal{A}$ and $\theta$ is a real parameter. Since the geometric stable process exhibits severe local singularities in its L\'evy measure, its transition semigroup lacks ultracontractivity, which invalidates classical methods for proving the differentiability. To overcome this obstacle, we use the compact embedding of the extended Dirichlet space into $L^2(\mu)$. As a primary application of this differentiability, we establish a large deviation principle for a positive continuous additive functional associated with the smooth measure $\mu$.

math.PR↗

arXiv2026

Classification and Metrization of Classes of Smooth measures

We classify the several classes of the set of smooth measures from the perspective of the denseness and the locality, and consider their relationships, in particular, that of the Kato class and Radon measures of finite energy integrals. We also introduce the Miyadera metric on the Dynkin class, and obtain the continuity of the Revuz correspondence.

math.PR↗

arXiv2026

A note on geometric α-stable processes and the existence of ground states for associated Schrödinger operators

In this paper, we establish the existence of transition density for geometric $α$-stable processes by using the property of self-decomposability--a fundamental concept in the theory of Lévy processes. In contrast to traditional and analytic methods that often rely on the $L^{1}$-integrability of the characteristic function, our approach is purely probabilistic and focuses on the structural regularity of the Lévy measure. As an application, we prove the existence of ground states for Schrödinger operators associated with recurrent geometric stable processes.

math.PR↗

arXiv2025

Smooth measures and positive continuous additive functionals attached to a compact nest

The relationship between smooth measures and positive continuous additive functionals is well known, and this correspondence is called the Revuz correspondence. We investigate the relationships between several types of convergence of smooth measures and convergence of positive continuous additive functionals, mainly focusing on a treatment of nests. We provide conditions under which convergence of additive functionals implies convergence of the corresponding smooth measures. Our results cover convergence of smooth measures that are not Radon, including nowhere Radon measures.

math.PR↗

arXiv2025

Harmonic functions for recurrent symmetric $α$-stable processeses with non-local perturbations

Let ${\mathbf M}$ be the recurrent symmetric (relativistic) $α$-stable process on ${\mathbb R}^d$. Let ${\mathcal H}^{μ+ F} (:= {\mathcal H} + μ+ F)$ be a Schrödinger type operator with local and non-local perturbations $μ$ and $F$. If $μ$ and $F$ satisfy suitable conditions associated with Kato class, we prove the existence of ground state for a Schrödinger type operator relating to ${\mathcal H}^{μ+ F}$. Furthermore, we prove the ground state becomes a probabilistically harmonic function of the Schrödinger operator generated by ${\mathcal H}^{μ+ F}$.

math.PR↗

arXiv2024

On a convergence of positive continuous additive functionals in terms of their smooth measures

A compactness of the Revuz map is established in the sense that the locally uniform convergence of a sequence of positive continuous additive functionals is derived in terms of their smooth measures. To this end, we first introduce a metric on the space of measures of finite energy integrals and show some structures of the metric. Then, we show the compactness and give some examples of positive continuous additive functionals that the convergence holds in terms of the associated smooth measures.

math.PR↗