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Takumu Ooi

Publications and source records attributed to Takumu Ooi.

9 recordsLinked to original sources
arXiv2026

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↗
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

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↗
arXiv2025

Homeomorphism of the Revuz correspondence for finite energy integrals

We provide necessary and sufficient conditions for the convergence of Revuz measures of finite energy integrals. More precisely, the Revuz map from the set of all smooth measures of finite energy integrals, equipped with the topology induced by the norm given by the sum of the Dirichlet form and the $L^2(m)$-norm, to the space of positive continuous additive functionals, equipped with the topology induced by the $L^2(\mathbb{P}_{m+κ+ν_0})$-norm with the local uniform topology, is a homeomorphism, where $m$ is the underlying measure, $κ$ is the killing measure of a Dirichlet form and $ν_0$ is an energy functional for the part that the process continuously escaping to the cemetery point.

math.PR↗
arXiv2024

Convergence of processes time-changed by Gaussian multiplicative chaos

As represented by the Liouville measure, Gaussian multiplicative chaos is a random measure constructed from a Gaussian field. Under certain technical assumptions, we prove the convergence of a process time-changed by Gaussian multiplicative chaos in the case the latter object is square integrable (the $L^2$-regime). As examples of the main result, we prove that, in the whole $L^2$-regime, the scaling limit of the Liouville simple random walk on $\mathbb{Z}^2$ is Liouville Brownian motion and, as $α\to 1$, Liouville $α$-stable processes on $\mathbb{R}$ converge weakly to the Liouville Cauchy process.

math.PR↗
arXiv2023

Dynkin games for Markov processes associated with semi-Dirichlet forms

We consider Dynkin games for Markov processes associated with semi-Dirichlet forms. Dynkin games are the optimal stopping games introduced as the models of zero-sum games by two players. We prove that the solution to the certain variational inequality with two obstacles is the equilibrium price of the Dynkin game. Moreover, we obtain the saddle point of the game.

math.PR↗
arXiv2022

Markov properties for Gaussian fields associated with Dirichlet forms

We prove the equivalence of the local property for an irreducible regular Dirichlet form and the Markov property for the Gaussian field associated with the Dirichlet form. Moreover we introduce a strong Markov property for Gaussian fields and present some sufficient conditions for this to hold.

math.PR↗
arXiv2020

Heat kernel estimates on spaces with varying dimension

We obtain sharp two-sided heat kernel estimates on spaces with varying dimension, in which two spaces of general dimension are connected at one point. On these spaces, if the dimensions of the two constituent parts are different, the volume doubling property fails with respect to the measure induced by the associated Lebesgue measures. Thus the parabolic Harnack inequalities fail and the heat kernels do not enjoy Aronson type estimates. Our estimates show that the on-diagonal estimates are independent of the dimensions of the two parts of the space for small time, whereas they depend on their transience or recurrence for large time. These are multidimensional version of a space considered by Z.-Q. Chen and S. Lou (Ann. Probab. 2019), in which a 1-dimensional space and a 2-dimensional space are connected at one point.

math.PR↗