arXiv subjects

Kazumasa Kuwada

Publications and source records attributed to Kazumasa Kuwada.

16 recordsLinked to original sources

arXiv2020

Radial processes for sub-Riemannian Brownian motions and applications

We study the radial part of sub-Riemannian Brownian motion in the context of totally geodesic foliations. Itô's formula is proved for the radial processes associated to Riemannian distances approximating the Riemannian one. We deduce very general stochastic completeness criteria for the sub-Riemannian Brownian motion. In the context of Sasakian foliations and H-type groups, one can push the analysis further, and taking advantage of the recently proved sub-Laplacian comparison theorems one can compare the radial processes for the sub-Riemannian distance to one-dimensional model diffusions. As a geometric application, we prove Cheng's type estimates for the Dirichlet eigenvalues of the sub-Riemannian metric balls, a result which seems to be new even in the Heisenberg group.

math.PR↗

arXiv2018

Monotonicity and rigidity of the W-entropy on RCD(0, N) spaces

By means of a space-time Wasserstein control, we show the monotonicity of the W-entropy functional in time along heat flows on possibly singular metric measure spaces with non-negative Ricci curvature and a finite upper bound of dimension in an appropriate sense. The associated rigidity result on the rate of dissipation of the W-entropy is also proved. These extend known results even on weighted Riemannian manifolds in some respects. In addition, we reveal that some singular spaces will exhibit the rigidity models while only the Euclidean space does in the class of smooth weighted Riemannian manifolds.

math.PR↗

arXiv2018

Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations

We develop a variational theory of geodesics for the canonical variation of the metric of a totally geodesic foliation. As a consequence, we obtain comparison theorems for the horizontal and vertical Laplacians. In the case of Sasakian foliations, we show that sharp horizontal and vertical comparison theorems for the sub-Riemannian distance may be obtained as a limit of horizontal and vertical comparison theorems for the Riemannian distances approximations.

math.DG↗

arXiv2017

Rigidity for the spectral gap on $RCD(K,\infty)$-spaces

We consider a rigidity problem for the spectral gap of the Laplacian on an $RCD(K,\infty)$-space (a metric measure space satisfying the Riemannian curvature-dimension condition) for positive $K$. For a weighted Riemannian manifold, Cheng--Zhou showed that the sharp spectral gap is achieved only when a $1$-dimensional Gaussian space is split off. This can be regarded as an infinite-dimensional counterpart to Obata's rigidity theorem. Generalizing to $RCD(K,\infty)$-spaces is not straightforward due to the lack of smooth structure and doubling condition. We employ the lift of an eigenfunction to the Wasserstein space and the theory of regular Lagrangian flows recently developed by Ambrosio--Trevisan to overcome this difficulty.

math.DG↗

arXiv2015

Equivalence between dimensional contractions in Wasserstein distance and the curvature-dimension condition

The curvature-dimension condition is a generalization of the Bochner inequality to weighted Riemannian manifolds and general metric measure spaces. It is now known to be equivalent to evolution variational inequalities for the heat semigroup, and quadratic Wasserstein distance contraction properties at different times. On the other hand, in a compact Riemannian manifold, it implies a same-time Wasserstein contraction property for this semigroup. In this work we generalize the latter result to metric measure spaces and more importantly prove the converse: contraction inequalities are equivalent to curvature-dimension conditions. Links with functional inequalities are also investigated.

math.PR↗

arXiv2014

A coupling of Brownian motions in the $\mathcal{L}_0$-geometry

Under a complete Ricci flow, we construct a coupling of two Brownian motion such that their $\mathcal{L}_0$-distance is a supermartingale. This recovers a result of Lott [J. Lott, Optimal transport and Perelman's reduced volume, Calc. Var. Partial Differential Equations 36 (2009), no. 1, 49--84.] on the monotonicity of $\mathcal{L}_0$-distance between heat distributions.

math.PR↗

arXiv2014

Space-time Wasserstein controls and Bakry-Ledoux type gradient estimates

The duality in Bakry-Émery's gradient estimates and Wasserstein controls for heat distributions is extended to that in refined estimates in a high generality. As a result, we find an equivalent condition to Bakry-Ledoux's refined gradient estimate involving an upper dimension bound. This new condition is described as a $L^2$-Wasserstein control for heat distributions at different times. The $L^p$-version of those estimates are studied on Riemannian manifolds via coupling method.

math.PR↗

arXiv2013

On the Equivalence of the Entropic Curvature-Dimension Condition and Bochner's Inequality on Metric Measure Spaces

We prove the equivalence of the curvature-dimension bounds of Lott-Sturm-Villani (via entropy and optimal transport) and of Bakry--Émery (via energy and Γ_2$-calculus) in complete generality for infinitesimally Hilbertian metric measure spaces. In particular, we establish the full Bochner inequality on such metric measure spaces. Moreover, we deduce new contraction bounds for the heat flow on Riemannian manifolds and on mms in terms of the $L^2$-Wasserstein distance.

math.DG↗

arXiv2012

Convergence of time-inhomogeneous geodesic random walks and its application to coupling methods

We study an approximation by time-discretized geodesic random walks of a diffusion process associated with a family of time-dependent metrics on manifolds. The condition we assume on the metrics is a natural time-inhomogeneous extension of lower Ricci curvature bounds. In particular, it includes the case of backward Ricci flow, and no further a priori curvature bound is required. As an application, we construct a coupling by reflection which yields a nice estimate of coupling time, and hence a gradient estimate for the associated semigroups.

math.PR↗

arXiv2012

Monotonicity of time-dependent transportation costs and coupling by reflection

Based on a study of the coupling by reflection of diffusion processes, a new monotonicity in time of a time-dependent transportation cost between heat distribution is shown under Bakry-Emery's curvature-dimension condition on a Riemannian manifold. The cost function comes from the total variation between heat distributions on spaceforms. As a corollary, we obtain a comparison theorem for the total variation between heat distributions. In addition, we show that our monotonicity is stable under the Gromov-Hausdorff convergence of the underlying space under a uniform curvature-dimension and diameter bound.

math.PR↗

arXiv2012

Heat flow on Alexandrov spaces

We prove that on compact Alexandrov spaces with curvature bounded below the gradient flow of the Dirichlet energy in the $L^2$-space produces the same evolution as the gradient flow of the relative entropy in the $L^2$-Wasserstein space. This means that the heat flow is well defined by either one of the two gradient flows. Combining properties of these flows, we are able to deduce the Lipschitz continuity of the heat kernel as well as Bakry-Émery gradient estimates and the $Γ_2$-condition. Our identification is established by purely metric means, unlike preceding results relying on PDE techniques. Our approach generalizes to the case of heat flow with drift.

math.DG↗

arXiv2010

Coupling of Brownian motions and Perelman's L-functional

We show that on a manifold whose Riemannian metric evolves under backwards Ricci flow two Brownian motions can be coupled in such a way that the expectation of their normalized L-distance is non-increasing. As an immediate corollary we obtain a new proof of a recent result of Topping (J. reine angew. Math. 636 (2009), 93-122), namely that the normalized L-transportation cost between two solutions of the heat equation is non-increasing as well.

math.PR↗

arXiv2010

Coupling by reflection of diffusion processes via discrete approximation under a backward Ricci flow

A coupling by reflection of a time-inhomogeneous diffusion process on a manifold are studied. The condition we assume is a natural time-inhomogeneous extension of lower Ricci curvature bounds. In particular, it includes the case of backward Ricci flow. As in time-homogeneous cases, our coupling provides a gradient estimate of the diffusion semigroup which yields the strong Feller property. To construct the coupling via discrete approximation, we establish the convergence in law of geodesic random walks as well as a uniform non-explosion type estimate.

math.PR↗

arXiv2009

Non-explosion of diffusion processes on manifolds with time-dependent metric

We study the problem of non-explosion of diffusion processes on a manifold with time-dependent Riemannian metric. In particular we obtain that Brownian motion cannot explode in finite time if the metric evolves under backwards Ricci flow. Our result makes it possible to remove the assumption of non-explosion in the pathwise contraction result established by Arnaudon, Coulibaly and Thalmaier (arXiv:0904.2762, to appear in Sem. Prob.). As an important tool which is of independent interest we derive an Ito formula for the distance from a fixed reference point, generalising a result of Kendall (Ann. Prob. 15 (1987), 1491--1500).

math.PR↗

arXiv2009

Duality on gradient estimates and Wasserstein controls

We establish a duality between L^p-Wasserstein control and L^q-gradient estimate in a general framework. Our result extends a known result for a heat flow on a Riemannian manifold. Especially, we can derive a Wasserstein control of a heat flow directly from the corresponding gradient estimate of the heat semigroup without using any other notion of lower curvature bound. By applying our result to a subelliptic heat flow on a Lie group, we obtain a coupling of heat distributions which carries a good control of their relative distance.

math.PR↗

arXiv2007

On uniqueness of maximal coupling for diffusion processes with a reflection

A maximal coupling of two diffusion processes makes two diffusion particles meet as early as possible. We study the uniqueness of maximal couplings under a sort of "reflection structure" which ensures the existence of such couplings. In this framework, the uniqueness in the class of Markovian couplings holds for the Brownian motion on a Riemannian manifold whereas it fails in more singular cases. We also prove that a Kendall-Cranston coupling is maximal under the reflection structure.

math.PR↗