arXiv subjects

Satoshi Ishiwata

Publications and source records attributed to Satoshi Ishiwata.

9 recordsLinked to original sources

arXiv2026

Hydrodynamic limit of exclusion processes with killing on weighted Riemannian manifolds via a graph discretization

In the present paper, we consider an exclusion process with killing on a proximity graph constructed from a partition of a geodesically complete weighted Riemannian manifold. We rigorously derive its hydrodynamic equation, which is governed by a heat equation with a killing potential term on the manifold. Within our graph discretization framework, we define an empirical density field for the exclusion process and prove its convergence, under an appropriate space-time scaling, to the unique bounded weak solution of the hydrodynamic equation, provided that the weighted manifold is stochastically complete. Combining local regularity for the parabolic equation with techniques from stochastic analysis on manifolds, we prove the uniqueness of the bounded weak solution and give the explicit representation of the solution in terms of the minimal Schr\"odinger kernel. Our result requires neither a global lower Ricci curvature bound nor global boundedness of the weight function.

math.PR↗

arXiv2023

A graph discretized approximation of semigroups for diffusion with drift and killing on a complete Riemannian manifold

In the present paper, we prove that the $C_{0}$-semigroup generated by a Schrödinger operator with drift on a complete Riemannian manifold is approximated by the discrete semigroups associated with a family of discrete time random walks with killing in a flow on a sequence of proximity graphs, which are constructed by partitions of the manifold. Furthermore, when the manifold is compact, we also obtain a quantitative error estimate of the convergence. Finally, we give examples of the partition of the manifold and the drift term on two typical manifolds: Euclidean spaces and model manifolds.

math.FA↗

arXiv2023

Poincaré constant on manifolds with ends

We obtain optimal estimates of the Poincaré constant of central balls on manifolds with finitely many ends. Surprisingly enough, the Poincaré constant is determined by the second largest end. The proof is based on the argument by Kusuoka-Stroock where the heat kernel estimates on the central balls play an essential role. For this purpose, we extend earlier heat kernel estimates obtained by the authors to a larger class of parabolic manifolds with ends.

math.DG↗

arXiv2020

Geometric analysis on manifolds with ends

In this survey article, we discuss some recent progress on geometric analysis on manifold with ends. In the final section, we construct manifolds with ends with oscillating volume functions which may turn out to have a different heat kernel estimates from those provided by known results.

math.DG↗

arXiv2019

Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part I

In the present paper, we study central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a point of view of discrete geometric analysis developed by Kotani and Sunada. We establish a semigroup CLT for a non-symmetric random walk on a nilpotent covering graph. Realizing the nilpotent covering graph into a nilpotent Lie group through a discrete harmonic map, we give a geometric characterization of the limit semigroup on the nilpotent Lie group. More precisely, we show that the limit semigroup is generated by the sub-Laplacian with a non-trivial drift on the nilpotent Lie group equipped with the Albanese metric. The drift term arises from the non-symmetry of the random walk and it vanishes when the random walk is symmetric. Furthermore, by imposing the "centered condition", we establish a functional CLT (i.e., Donsker-type invariance principle) in a Hoelder space over the nilpotent Lie group. The functional CLT is extended to the case where the realization is not necessarily harmonic. We also obtain an explicit representation of the limiting diffusion process on the nilpotent Lie group and discuss a relation with rough path theory. Finally, we give several examples of random walks on nilpotent covering graphs with explicit computations.

math.PR↗

arXiv2019

Central limit theorems for non-symmetric random walks on nilpotent covering graphs: Part II

In the present paper, as a continuation of our preceding paper [10], we study another kind of central limit theorems (CLTs) for non-symmetric random walks on nilpotent covering graphs from a viewpoint of discrete geometric analysis developed by Kotani and Sunada. We introduce a one-parameter family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We first prove a semigroup CLT for the family of random walks by realizing the nilpotent covering graph into a nilpotent Lie group via discrete harmonic maps. The limiting diffusion semigroup is generated by the homogenized sub-Laplacian with a constant drift of the asymptotic direction on the nilpotent Lie group, which is equipped with the Albanese metric associated with the symmetrized random walk. We next prove a functional CLT (i.e., Donsker-type invariance principle) in a Holder space over the nilpotent Lie group by combining the semigroup CLT, standard martingale techniques, and a novel pathwise argument inspired by rough path theory. Applying the corrector method, we finally extend these CLTs to the case where the realizations are not necessarily harmonic.

math.PR↗

arXiv2016

Heat kernel estimates on connected sums of parabolic manifolds

We obtain matching two sided estimates of the heat kernel on a connected sum of parabolic manifolds, each of them satisfying the Li-Yau estimate. The key result is the on-diagonal upper bound of the heat kernel at a central point. Contrary to the nonparabolic case (which was settled in [15]), the on-diagonal behavior of the heat kernel in our case is determined by the end with the maximal volume growth function. As examples, we give explicit heat kernel bounds on the connected sums $R^2#R^2$ and $R^1#R^2$ where $R^1 = R_+\timesS^1$.

math.PR↗

arXiv2015

Long time asymptotics of non-symmetric random walks on crystal lattices

In the present paper, we study long time asymptotics of non-symmetric random walks on crystal lattices from a view point of discrete geometric analysis due to Kotani and Sunada [11, 23]. We observe that the Euclidean metric associated with the standard realization of the crystal lattice, called the Albanese metric, naturally appears in the asymptotics. In the former half of the present paper, we establish two kinds of (functional) central limit theorems for random walks. We first show that the Brownian motion on the Euclidean space with the Albanese metric appears as the scaling limit of the usual central limit theorem for the random walk. Next we introduce a family of random walks which interpolates between the original non-symmetric random walk and the symmetrized one. We then capture the Brownian motion with a constant drift of the asymptotic direction on the Euclidean space with the Albanese metric associated with the symmetrized random walk through another kind of central limit theorem for the family of random walks. In the latter half of the present paper, we give a spectral geometric proof of the asymptotic expansion of the $n$-step transition probability for the non-symmetric random walk. This asymptotic expansion is a refinement of the local central limit theorem obtained by Sunada [22] and is a generalization of the result in [11] for symmetric random walks on crystal lattices to non-symmetric cases.

math.PR↗

arXiv2012

An explicit effect of non-symmetry of random walks on the triangular lattice

In the present paper, we study an explicit effect of non-symmetry on asymptotics of the $n$-step transition probability as $n\rightarrow \infty$ for a class of non-symmetric random walks on the triangular lattice. Realizing the triangular lattice into $\mathbb{R}^2$ appropriately, we observe that the Euclidean distance in $\mathbb{R}^2$ naturally appears in the asymptotics. We characterize this realization from a geometric view point of Kotani-Sunada's standard realization of crystal lattices. As a corollary of the main theorem, we prove that the transition semigroup generated by the non-symmetric random walk approximates the heat semigroup generated by the usual Brownian motion on $\mathbb{R}^2$.

math.PR↗