Furstenberg (1971) and Lyons and Sullivan (1984) have shown how to discretize harmonic functions on a Riemannian manifold M whose Brownian motion satisfies a certain recurrence property called ∗-recurrence. We study analogues of this discretization for tensor fields which are harmonic in the sense of the covariant Laplacian. We show that, under certain restrictions on the holonomy of the connection, the lifted diffusion on the orthonormal frame bundle has the same ∗-recurrence property as the original Brownian motion. This observation leads us to introduce a technique we call scalarization which reduces the problem of discretization for a tensor field to that of ordinary harmonic functions.

Furstenberg（1971）和Lyons和Sullivan（1984）展示了如何离散黎曼流形M上的调和函数，该流形的布朗运动满足一定的递归特性，即∗递归。我们研究了张量场的离散化的类似物，张量场在协变拉普拉斯算子的意义上是谐波的。我们表明，在对连接整体性的某些限制下，正交框架束上的提升扩散具有与原始布朗运动相同的*重复性质。这种观察导致我们引入了一种称为标量化的技术，该技术将张量场的离散化问题减少到普通谐波函数的离散化问题。