arXiv subjects

Ryoichiro Noda

Publications and source records attributed to Ryoichiro Noda.

8 recordsLinked to original sources

arXiv2026

Exact Hausdorff measures and fine geometry of collisions of independent Markov processes

We study the fine geometry of collision measures associated with independent Markov processes on a common metric measure space. These measures are natural space-time random measures encoding collision times and collision points. Under small-scale Ahlfors regularity and standard short-time heat-kernel estimates, we determine the local dimensions of their temporal and spatial marginals, the Hausdorff dimensions of their supports and of the collision sets themselves, and logarithmic limsup laws for their local masses at typical collisions and at exceptional thick collisions. Through this analysis, we identify exact Hausdorff measure functions for the collision-time sets and, in a natural regime, for the collision-point sets. Moreover, we prove that the corresponding Hausdorff measures are comparable, uniformly over all Borel subsets, to the temporal and spatial marginals of the collision measures. This provides a broad solution to the Hausdorff-measure part of an open question posed by Xiao more than two decades ago. Our framework covers symmetric stable processes, canonical diffusions on affine nested fractals such as the Sierpiński gasket, and stable-like jump processes on $d$-sets.

math.PR↗

arXiv2026

Continuity of the Revuz correspondence under the absolute continuity condition

In this paper, we consider standard processes that admit dual processes and satisfy the absolute continuity condition, i.e., processes possess transition densities. For such processes, the Revuz correspondence relates positive continuous additive functionals (PCAFs) to so-called smooth measures. We show the continuity of this correspondence. Specifically, we show that if the $1$-potentials of smooth measures converge (locally) uniformly as functions, then the associated PCAFs converge. This result is derived by directly estimating the distance between the PCAFs in terms of the distance between the $1$-potentials of the associated smooth measures. Furthermore, in cases where the transition density is jointly continuous, we present sufficient conditions for the convergence of $1$-potentials based on the weak or vague convergence of smooth measures. The framework in this paper contains the class of symmetric Hunt processes that are associated with regular Dirichlet forms and satisfy the absolute continuity condition.

math.PR↗

arXiv2026

Generalized Kac's moment formula for positive continuous additive functionals of Markov processes

We establish a formula for moments of certain random variables involving positive continuous additive functionals (PCAFs) of standard processes which have absolutely continuous transition functions and are in duality with standard processes with absolutely continuous transition functions,generalizing the classical Kac's moment formula. In particular, all our results are applicable to the more familiar case of symmetric Hunt processes which are associated with regular Dirichlet forms and have absolutely continuous transition functions.

math.PR↗

arXiv2025

Scaling limits of discrete-time Markov chains and their local times on electrical networks

We establish that if a sequence of electrical networks equipped with conductance measures converges in the local Gromov--Hausdorff-vague topology and satisfies certain non-explosion and metric-entropy conditions,then the sequence of associated discrete-time Markov chains and their local times also converges. This result applies to many examples, such as critical Galton--Watson trees conditioned on size, uniform spanning trees, random recursive fractals, the critical Erdős--Rényi random graph, the configuration model, and the random conductance model on fractals.To obtain the convergence result, we characterize and study extended Dirichlet spaces associated with resistance forms, and we study traces of electrical networks.

math.PR↗

arXiv2025

Convergence of space-time occupation measures of stochastic processes and its application to collisions

We introduce a new perspective on positive continuous additive functionals (PCAFs) of Markov processes, which we call space--time occupation measures (STOMs). This notion provides a natural generalization of classical occupation times and occupation measures, and offers a unified framework for studying their convergence. We analyze STOMs via so-called smooth measures associated with PCAFs through the Revuz correspondence. We establish that if the underlying spaces, the processes living on them, their heat kernels, and the associated smooth measures converge, and if the corresponding potentials of these measures satisfy a uniform decay condition, then the associated PCAFs and STOMs also converge in suitable Gromov--Hausdorff-type topologies. We then apply this framework to the analysis of collisions of independent stochastic processes. Specifically, by exploiting the STOM formulation, we introduce the notion of collision measures, which record both the collision sites and times of two processes, and prove general convergence theorems for these measures. The abstract results are further specialized to random walks on electrical networks via the theory of resistance metric spaces, leading to concrete scaling limits for collision measures of random walks on critical random graphs, such as critical Galton--Watson trees, critical Erdős--Rényi random graphs, and the uniform spanning tree.

math.PR↗

arXiv2025

Metrization of Gromov-Hausdorff-type topologies on boundedly-compact metric spaces

We present a new general framework for metrization of Gromov-Hausdorff-type topologies on non-compact metric spaces. We also give easy-to-check conditions for separability and completeness and hence the measure theoretic requirements are provided to study convergence of random spaces with additional random objects. In particular, our framework enables us to define a metric inducing a suitable Gromov-Hausdorff-type topology on the space of rooted boundedly-compact metric spaces with laws of stochastic processes and/or random fields, which was not clear how to do in previous frameworks. In addition to general theory, this paper includes several examples of Gromov-Hausdorff-type topologies, verifying that classical examples such as the Gromov-Hausdorff topology and the Gromov-Hausdorff-Prohorov topology are contained within our framework.

math.MG↗

arXiv2024

Aging and sub-aging for Bouchaud trap models on resistance metric spaces

In this paper, we prove that if a sequence of electrical networks converges in the local Gromov-Hausdorff topology and satisfies a non-explosion condition, then the associated Bouchaud trap models (BTMs) also converge and exhibit aging. Moreover, when local structures of electrical networks converge, we prove sub-aging. Our results are applicable to a wide class of low-dimensional graphs, including the two-dimensional Sierpiński gasket, critical Galton-Watson trees, and the critical Erdős-Rényi random graph. The proof consists of two main steps: Polish metrization of the vague-and-point-process topology and showing the precompactness of transition densities of BTMs.

math.PR↗

arXiv2024

Convergence of local times of stochastic processes associated with resistance forms

In this paper, it is shown that if a sequence of resistance metric spaces equipped with measures converges with respect to the local Gromov-Hausdorff-vague topology, and certain non-explosion and metric-entropy conditions are satisfied, then the associated stochastic processes and their local times also converge. The metric-entropy condition can be checked by applying volume estimates of balls. Whilst similar results have been proved previously, the approach of this article is more widely applicable. Indeed, we recover various known conclusions for scaling limits of some deterministic self-similar fractal graphs, critical Galton-Watson trees, the critical Erdős-Rényi random graph and the configuration model (in the latter two cases, we prove for the first time the convergence of the models with respect to the resistance metric and also, for the configuration model, we overcome an error in the existing proof of local time convergence). Moreover, we derive new ones for scaling limits of uniform spanning trees and random recursive fractals. The metric-entropy condition also implies convergence of associated Gaussian processes.

math.PR↗