AI 中文总结
本文针对欧几里得空间子流形切丛上的不连续鞅，建立切丛上不连续半鞅的伊藤微积分，推导调和映射的导数相关性质及均值、导数公式，拓展了随机分析与几何的相关理论。
AI 中文摘要
本文研究欧几里得空间中子流形的切丛上的不连续鞅。首先，我们在切丛上引入联络规则，并建立切丛上不连续半鞅的伊藤微积分。接着，我们关注关于非局部狄利克雷形式的调和映射，证明无穷小对称下调和映射的导数会在切丛上诱导出不连续鞅，该过程可视为沿像鞅的随机雅可比场。我们还引入了带投影跳跃的黎曼子流形上càdlàg半鞅的切向量随机平行运输，利用该平行运输，我们得到了调和映射微分的均值性质，其包含通过第二基本形式表示的跳跃部分。此外，我们还得到了紧致黎曼流形上带布朗分量的各向同性Lévy过程的调和映射导数公式。
英文摘要
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.
Comments39 pages
