arXiv subjects

Shin-ichi Ohta

Publications and source records attributed to Shin-ichi Ohta.

At least 19 recordsLinked to original sources

arXiv2026

Brownian motion in Minkowski normed spaces

A Minkowski normed space is the Euclidean space equipped with a (possibly asymmetric) uniformly convex and smooth norm, forming a particular class of Finsler manifolds. We construct a stochastic process with one-dimensional time marginal densities given by the fundamental solution to the nonlinear Finsler heat equation in Minkowski normed spaces. This process is constructed as a solution to a singular McKean--Vlasov stochastic differential equation and constitutes a nonlinear Markov process in the sense of McKean. Furthermore, we show that solutions to this stochastic differential equation are pathwise unique, and thus probabilistically strong solutions, though the equation has singular coefficients beyond the subcritical regime. Since our construction is a natural extension of the construction of standard Brownian motion from the standard heat kernel, we call this process \emph{Brownian motion in Minkowski normed spaces.} To the best of our knowledge, this is the first construction of stochastic processes associated with nonlinear heat equation in Finslerian spaces.

math.PR↗

arXiv2025

Concavity of spacetimes

Motivated by recent breathtaking progress in the synthetic study of Lorentzian geometry, we investigate the local concavity of time separation functions on Finsler spacetimes as a Lorentzian counterpart to Busemann's convexity in metric geometry. We show that a Berwald spacetime is locally concave if and only if its flag curvature is nonnegative in timelike directions. We also give another characterization of nonnegative flag curvature by the convexity of future (or past) capsules, inspired by Krist\'aly--Kozma's result in the positive definite case. These characterizations are new even for Lorentzian manifolds.

math.DG↗

arXiv2025

Analysis on asymmetric metric measure spaces: $q$-heat flow, $q$-Laplacian and Sobolev spaces

We investigate geometric analysis on metric measure spaces equipped with asymmetric distance functions. Extending concepts from the symmetric case, we introduce upper gradients and corresponding $L^q$-energy functionals as well as $q$-Laplacian in the asymmetric setting. Along the lines of gradient flow theory, we then study $q$-heat flow as the $L^2$-gradient flow for $L^q$-energy. We also study the Sobolev spaces over asymmetric metric measure spaces, extending some results of Ambrosio--Gigli--Savaré to asymmetric distances. A wide class of asymmetric metric measure spaces is provided by irreversible Finsler manifolds, which serve to construct various model examples by pointing out genuine differences between the symmetric and asymmetric settings.

math.DG↗

arXiv2025

How does the contraction property fail for convex functions on normed spaces?

On Euclidean and Hilbert spaces, Riemannian manifolds, and CAT$(0)$-spaces, gradient flows of convex functions are known to satisfy the contraction property, which plays a fundamental role in optimization theory and possesses fruitful analytic and geometric applications. On (non-inner product) normed spaces, however, gradient flows of convex functions do not satisfy the contraction property. We give a detailed proof of this characterization of inner products, and discuss a possible form of a weaker contraction property on normed spaces.

math.OC↗

arXiv2025

Integral Varadhan formula for non-linear heat flow

We prove the integral Varadhan short-time formula for non-linear heat flow on measured Finsler manifolds. To the best of the authors' knowledge, this is the first result establishing a Varadhan-type formula for non-linear semigroups. We do not assume the reversibility of the metric, and the distance function can be asymmetric. In this generality, we reveal that the probabilistic interpretation is well-suited for our formula; the probability that a particle starting from a set $A$ can be found in another set $B$ describes the distance from $A$ to $B$. One side of the estimates (the upper bound of the probability) is also established in the nonsmooth setting of infinitesimally strictly convex metric measure spaces satisfying the local Sobolev-to-Lipschitz property.

math.PR↗

arXiv2025

Discrete-time gradient flows for unbounded convex functions on Gromov hyperbolic spaces

In proper, geodesic Gromov hyperbolic spaces, we investigate discrete-time gradient flows via the proximal point algorithm for unbounded Lipschitz convex functions. Assuming that the target convex function has negative asymptotic slope along some ray (thus unbounded below), we first prove the uniqueness of such a negative direction in the boundary at infinity. Then, we show that the discrete-time gradient flow from an arbitrary initial point diverges to that unique direction of negative asymptotic slope. This is inspired by and generalizes results of Karlsson-Margulis and Hirai-Sakabe on nonpositively curved spaces and a result of Karlsson concerning semi-contractions on Gromov hyperbolic spaces. We also give an estimate of the rate of convergence based on a contraction property for the proximal (resolvent) operator established in our previous work.

math.OC↗

arXiv2025

Discrete-time gradient flows in Gromov hyperbolic spaces

We investigate fundamental properties of the proximal point algorithm for Lipschitz convex functions on (proper, geodesic) Gromov hyperbolic spaces. We show that the proximal point algorithm from an arbitrary initial point can find a point close to a minimizer of the function. Moreover, we establish contraction estimates (akin to trees) for the proximal (resolvent) operator. Our results can be applied to small perturbations of trees.

math.OC↗

arXiv2024

Splitting theorems for weighted Finsler spacetimes via the $p$-d'Alembertian: beyond the Berwald case

A timelike splitting theorem for Finsler spacetimes was previously established by the third author, in collaboration with Lu and Minguzzi, under relatively strong hypotheses, including the Berwald condition. This contrasts with the more general results known for positive definite Finsler manifolds. In this article, we employ a recently developed strategy for proving timelike splitting theorems using the elliptic $p$-d'Alembertian. This approach, pioneered by Braun, Gigli, McCann, S\"amann, and the second author, allows us to remove the restrictive assumptions of the earlier splitting theorem. For timelike geodesically complete Finsler spacetimes, we establish a diffeomorphic splitting. In the specific case of Berwald spacetimes, we show that the Busemann function generates a group of isometries via translations. Furthermore, for Berwald spacetimes, we extend these splitting theorems by replacing the assumption of timelike geodesic completeness with global hyperbolicity. Our results encompass and generalize the timelike splitting theorems for weighted Lorentzian manifolds previously obtained by Case and Woolgar-Wylie.

math.DG↗

arXiv2024

A generalization of Gr\"unbaum's inequality in RCD$(0,N)$-spaces

We generalize Gr\"unbaum's classical inequality in convex geometry to curved spaces with nonnegative Ricci curvature, precisely, to $\mathrm{RCD}(0,N)$-spaces with $N \in (1,\infty)$ as well as weighted Riemannian manifolds of $\mathrm{Ric}_N \ge 0$ for $N \in (-\infty,-1) \cup \{\infty\}$. Our formulation makes use of the isometric splitting theorem; given a convex set $\Omega$ and the Busemann function associated with any straight line, the volume of the intersection of $\Omega$ and any sublevel set of the Busemann function that contains a barycenter of $\Omega$ is bounded from below in terms of $N$. We also extend this inequality beyond uniform distributions on convex sets. Moreover, we establish some rigidity results by using the localization method, and the stability problem is also studied.

math.MG↗

arXiv2024

Barycenters and a law of large numbers in Gromov hyperbolic spaces

We investigate barycenters of probability measures on Gromov hyperbolic spaces, toward development of convex optimization in this class of metric spaces. We establish a contraction property (the Wasserstein distance between probability measures provides an upper bound of the distance between their barycenters), a deterministic approximation of barycenters of uniform distributions on finite points, and a kind of law of large numbers. These generalize the corresponding results on CAT(0)-spaces, up to additional terms depending on the hyperbolicity constant.

math.MG↗

arXiv2023

Gradient flows in asymmetric metric spaces and applications

This paper is devoted to the investigation of gradient flows in asymmetric metric spaces (for example, irreversible Finsler manifolds and Minkowski normed spaces) by means of discrete approximation. We study basic properties of curves and upper gradients in asymmetric metric spaces, and establish the existence of a curve of maximal slope, which is regarded as a gradient curve in the non-smooth setting. { Introducing} a natural convexity assumption on the potential function, { which is called the $(p,λ)$-convexity,} we also obtain some regularizing effects on the asymptotic behavior of curves of maximal slope. Applications include several existence results for gradient flows in Finsler manifolds, doubly nonlinear differential evolution equations on { infinite-dimensional Funk spaces}, and heat flow on compact Finsler manifolds.

math.DG↗

arXiv2023

Optimal transport and timelike lower Ricci curvature bounds on Finsler spacetimes

We prove that a Finsler spacetime endowed with a smooth reference measure whose induced weighted Ricci curvature $\smash{\mathrm{Ric}_N}$ is bounded from below by a real number $K$ in every timelike direction satisfies the timelike curvature-dimension condition $\smash{\mathrm{TCD}_q(K,N)}$ for all $q\in (0,1)$. A nonpositive-dimensional version ($N \le 0$) of this result is also shown. Our discussion is based on the solvability of the Monge problem with respect to the $q$-Lorentz-Wasserstein distance as well as the characterization of $q$-geodesics of probability measures. One consequence of our work is the sharp timelike Brunn-Minkowski inequality in the Lorentz-Finsler case.

math.DG↗

arXiv2022

Geometry of weighted Lorentz-Finsler manifolds II: A splitting theorem

We show an analogue of the Lorentzian splitting theorem for weighted Lorentz-Finsler manifolds: If a weighted Berwald spacetime of nonnegative weighted Ricci curvature satisfies certain completeness and metrizability conditions and includes a timelike straight line, then it necessarily admits a one-dimensional family of isometric translations generated by the gradient vector field of a Busemann function. Moreover, our formulation in terms of the $ε$-range introduced in our previous work enables us to unify the previously known splitting theorems for weighted Lorentzian manifolds by Case and Woolgar-Wylie into a single framework.

math.DG↗

arXiv2022

Quantitative estimates for the Bakry-Ledoux isoperimetric inequality. II

Concerning quantitative isoperimetry for a weighted Riemannian manifold satisfying $\mathrm{Ric}_{\infty} \ge 1$, we give an $L^1$-estimate exhibiting that the push-forward of the reference measure by the guiding function (arising from the needle decomposition) is close to the Gaussian measure. We also show $L^p$- and $W_2$-estimates in the $1$-dimensional case.

math.DG↗

arXiv2022

Nonlinear geometric analysis on Finsler manifolds

This is a survey article on recent progress of comparison geometry and geometric analysis on Finsler manifolds of weighted Ricci curvature bounded below. Our purpose is two-fold: Give a concise and geometric review on the birth of weighted Ricci curvature and its applications; Explain recent results from a nonlinear analogue of the $Γ$-calculus based on the Bochner inequality. In the latter we discuss some gradient estimates, functional inequalities, and isoperimetric inequalities.

math.DG↗

arXiv2022

Vanishing S-curvature of Randers spaces

We give a necessary and sufficient condition on a Randers space for the existence of a measure for which Shen's S-curvature vanishes everywhere. Moreover, such a measure coincides with the Busemann-Hausdorff measure up to a constant multiplication.

math.DG↗

arXiv2022

Some functional inequalities on non-reversible Finsler manifolds

We continue our study of geometric analysis on (possibly non-reversible) Finsler manifolds, based on the Bochner inequality established by the author and Sturm. Following the approach of the $Γ$-calculus a la Bakry et al, we show the dimensional versions of the Poincare--Lichnerowicz inequality, the logarithmic Sobolev inequality, and the Sobolev inequality. In the reversible case, these inequalities were obtained by Cavalletti--Mondino in the framework of curvature-dimension condition by means of the localization method. We show that the same (sharp) estimates hold also for non-reversible metrics.

math.DG↗

arXiv2022

Comparison theorems on weighted Finsler manifolds and spacetimes with $ε$-range

We establish the Bonnet-Myers theorem, Laplacian comparison theorem, and Bishop-Gromov volume comparison theorem for weighted Finsler manifolds as well as weighted Finsler spacetimes, of weighted Ricci curvature bounded below by using the weight function. These comparison theorems are formulated with $ε$-range introduced in our previous paper, that provides a natural viewpoint of interpolating weighted Ricci curvature conditions of different effective dimensions. Some of our results are new even for weighted Riemannian manifolds and generalize comparison theorems of Wylie-Yeroshkin and Kuwae-Li.

math.DG↗