arXiv subjects

Fumiya Okazaki

Publications and source records attributed to Fumiya Okazaki.

8 recordsLinked to original sources

arXiv2026

On sequences of martingales with jumps on Riemannian submanifolds

In this article, we investigate sequences of discontinuous martingales on submanifolds of higher-dimensional Euclidean space. Those sequences naturally arise when we deal with a sequence of harmonic maps with respect to non-local Dirichlet forms, such as fractional harmonic maps. We prove that the semimartingale topology is equivalent to the topology of locally uniform convergence in probability on the space of discontinuous martingales on manifolds with bounded jumps. In particular, we show that the limit of any sequence of discontinuous martingales on a compact Riemannian manifold, with respect to the topology of locally uniform convergence in probability, is a martingale on the manifold. As an application, we prove that for sequences of quasi-harmonic maps on an open set with respect to a non-local Dirichlet form, local $L^{\infty}$-convergence in the open set with pointwise convergence q.e. on the whole space implies strong local convergence in the Dirichlet space and the limit preserves harmonicity.

math.PR↗

arXiv2026

Stochastic Jacobi fields along discontinuous martingales on Riemannian submanifolds

In this article, we consider discontinuous martingales on tangent bundles over submanifolds of Euclidean space. First, we introduce a connection rule on tangent bundles and establish the Itô calculus for discontinuous semimartingales on tangent bundles. Then we focus on harmonic maps with respect to non-local Dirichlet forms and show that the derivative of harmonic maps along infinitesimal symmetries induces discontinuous martingales on tangent bundles. This process may be viewed as a stochastic Jacobi field along the image martingale. We also introduce the stochastic parallel transport of tangent vectors along càdlàg semimartingales on Riemannian submanifolds with projected jumps. Using the parallel transport, we obtain the mean-value property for the differential of harmonic maps involving a jump part expressed through the second fundamental form. We also obtain a derivative formula for harmonic maps for isotropic Lévy processes with a Brownian component on compact Riemannian manifolds.

math.PR↗

arXiv2026

Topology optimization concerning the mass distribution via filtered gradient flows on the Wasserstein space

In this article, we formulate topology optimization problems concerning the mass distribution as minimization problems for functionals on the Wasserstein space. We relax optimization problems regarding non-convex objective functions on the Wasserstein space by using the Neumann heat semigroup and prove the existence of minimizers of relaxed problems. Furthermore, we introduce the filtered Wasserstein gradient flow and derive the error estimate between the original Wasserstein gradient flow and the filtered one in terms of the Wasserstein distance. We also construct a candidate for the optimal mass distribution for a given fixed total mass and simultaneously obtain the shape of the material by the numerical calculation of filtered Wasserstein gradient flows.

math.OC↗

arXiv2025

On sensitivities regarding shape and topology optimization as derivatives on Wasserstein spaces

In this paper, we apply the framework of optimal transport to the formulation of optimal design problems. By considering the Wasserstein space as a set of design variables, we associate each probability measure with a shape configuration of a material in some ways. In particular, we focus on connections between differentials on the Wasserstein space and sensitivities in the standard setting of shape and topology optimization in order to regard the optimization procedure of those problems as gradient flows on the Wasserstein space.

math.OC↗

arXiv2024

Horizontal $Δ$-semimartingales on orthonormal frame bundles

In this article, we deal with stochastic horizontal lifts and anti-developments of semimartingales with jumps on complete and connected Riemannian manifolds without any assumption for their curvatures. We prove two one-to-one correspondences among some classes of discontinuous semimartingales on Riemannian manifolds, orthonormal frame bundles and Euclidean spaces by using the stochastic differential geometry with jumps introduced by Cohen (1996). Both of these two results are extension of the one shown in Pontier-Estrade (1992). The first result is the correspondence in the case where jumps of semimartingales are regarded as initial velocities of geodesics which are not necessarily minimal. In the second result, we also established the correspondence in the situation where jumps of semimartingales are given by connection rules, but we impose the condition that the jumps of semimartingales are small. The latter result enables us to construct martingales for a given connection rule with small jumps on any compact manifold from local martingales on a Euclidean space through horizontal semimartingales on orthonormal semimartingales.

math.PR↗

arXiv2024

Probabilistic characterization of weakly harmonic maps with respect to non-local Dirichlet forms

We characterize weakly harmonic maps with respect to non-local Dirichlet forms by Markov processes and martingales. In particular, we can obtain discontinuous martingales on Riemannian manifolds from the image of symmetric stable processes under fractional harmonic maps in a weak sense. Based on this characterization, we also consider the continuity of weakly harmonic maps along the paths of Markov processes and describe the condition for the continuity of harmonic maps by quadratic variations of martingales in some situations containing cases of energy minimizing maps.

math.PR↗

arXiv2023

Convergence of martingales with jumps on submanifolds of Euclidean spaces and its applications to harmonic maps

Martingales with jumps on Riemannian manifolds and harmonic maps with respect to Markov processes are discussed in this paper. Discontinuous martingales on manifolds were introduced in Picard (1991). We obtain results about the convergence of martingales with finite quadratic variations on Riemannian submanifolds of higher dimensional Euclidean space as $t\to \infty$ and $t\to 0$. Furthermore we apply the result about martingales with jumps on submanifolds to harmonic maps with respect to Markov processes such as fractional harmonic maps.

math.PR↗

arXiv2018

Jointly learning relevant subgraph patterns and nonlinear models of their indicators

Classification and regression in which the inputs are graphs of arbitrary size and shape have been paid attention in various fields such as computational chemistry and bioinformatics. Subgraph indicators are often used as the most fundamental features, but the number of possible subgraph patterns are intractably large due to the combinatorial explosion. We propose a novel efficient algorithm to jointly learn relevant subgraph patterns and nonlinear models of their indicators. Previous methods for such joint learning of subgraph features and models are based on search for single best subgraph features with specific pruning and boosting procedures of adding their indicators one by one, which result in linear models of subgraph indicators. In contrast, the proposed approach is based on directly learning regression trees for graph inputs using a newly derived bound of the total sum of squares for data partitions by a given subgraph feature, and thus can learn nonlinear models through standard gradient boosting. An illustrative example we call the Graph-XOR problem to consider nonlinearity, numerical experiments with real datasets, and scalability comparisons to naive approaches using explicit pattern enumeration are also presented.

cs.LG↗