论文标题
通过连接的光滑流线上的Lévy过程
Lévy processes on smooth manifolds with a connection
论文作者
论文摘要
我们在平滑的歧管$ m $上定义了莱维过程，并连接连接，以预测Marcus随机微分方程的解决方案，这是由$ M $的单$ $ m $的，这是由欧几里得空间上的自由主义lévy流程驱动的。在Riemannian歧管上，我们的定义（具有Levi-Civita连接）概括了Brownian Motion的Eells-Elworthy-Malliavin构造，并扩展了Applebaum和Estrade引入的各向同性Lévy过程的类别[AE00]。在具有支柱指数图的谎言组上，我们的定义（具有左右连接）与（左）lévy过程的经典定义相吻合。 我们的主要定理通过其发电机上的$ m $来表征Lévy过程的类别，从而推广了Laplace-Beltrami操作员在Riemannian歧管上产生Brownian Motion的事实。它的证明需要逐步构造随机水平升力和不连续的半卫星的反开发，从而导致Pontier和Estrade [PE92]的概括，以平滑具有不同点之间非独特的大地测量的光滑歧管。
We define a Lévy process on a smooth manifold $M$ with a connection as a projection of a solution of a Marcus stochastic differential equation on a holonomy bundle of $M$, driven by a holonomy-invariant Lévy process on a Euclidean space. On a Riemannian manifold, our definition (with Levi-Civita connection) generalizes the Eells-Elworthy-Malliavin construction of the Brownian motion and extends the class of isotropic Lévy process introduced in Applebaum and Estrade [AE00]. On a Lie group with a surjective exponential map, our definition (with left-invariant connection) coincides with the classical definition of a (left) Lévy process given in terms of its increments. Our main theorem characterizes the class of Lévy processes via their generators on $M$, generalizing the fact that the Laplace-Beltrami operator generates Brownian motion on a Riemannian manifold. Its proof requires a path-wise construction of the stochastic horizontal lift and anti-development of a discontinuous semimartingale, leading to a generalization of Pontier and Estrade [PE92] to smooth manifolds with non-unique geodesics between distinct points.
