Language：English  

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.

arXiv

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Other Link： https://arxiv.org/pdf/2608.15379v1

Authorship：Lead author   Language：English  

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.

arXiv

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Other Link： https://arxiv.org/pdf/2601.14332v1

Authorship：Lead author   Language：English  

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.

arXiv

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Other Link： http://arxiv.org/pdf/2411.12234v1

Language：English  

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.

arXiv

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Other Link： http://arxiv.org/pdf/2105.04824v2

Language：English  

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.

arXiv

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Other Link： https://arxiv.org/pdf/2409.17118v2

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