Let Q be a differential operator of order \(\le 1\) on a complex metric vector bundle \(\mathscr {E}\rightarrow \mathscr {M}\) with metric connection \(\nabla \) over a possibly noncompact Riemannian manifold \(\mathscr {M}\). Under very mild regularity assumptions on Q that guarantee that \(\nabla ^{\dagger }\nabla /2+Q\) canonically induces a holomorphic semigroup \(\mathrm {e}^{-zH^{\nabla }_{Q}}\) in \(\Gamma _{L^2}(\mathscr {M},\mathscr {E})\) (where z runs through a complex sector which contains \([0,\infty )\)), we prove an explicit Feynman–Kac type formula for \(\mathrm {e}^{-tH^{\nabla }_{Q}}\), \(t>0\), generalizing the standard self-adjoint theory where Q is a self-adjoint zeroth order operator. For compact \(\mathscr {M}\)’s we combine this formula with Berezin integration to derive a Feynman–Kac type formula for an operator trace of the form

where \(V,\widetilde{V}\) are of zeroth order and P is of order \(\le 1\). These formulae are then used to obtain a probabilistic representations of the lower order terms of the equivariant Chern character (a differential graded extension of the JLO-cocycle) of a compact even-dimensional Riemannian spin manifold, which in combination with cyclic homology play a crucial role in the context of the Duistermaat–Heckmann localization formula on the loop space of such a manifold.

令Q是一个复杂度量向量丛\(\mathscr {E}\rightarrow \mathscr {M}\)上的阶微分算子\(\le 1\) ，其度量连接\(\nabla \)黎曼流形\(\mathscr {M}\)。在Q的非常温和的正则性假设下，保证\(\nabla ^{\dagger }\nabla /2+Q\)规范地导出一个全纯半群\(\mathrm {e}^{-zH^{\nabla }_{ Q}}\)在\(\Gamma _{L^2}(\mathscr {M},\mathscr {E})\)中（其中z穿过包含\([0,\infty )\的复杂扇区))，我们证明了\(\mathrm {e}^{-tH^{\nabla }_{Q}}\) , \(t>0\)的显式 Feynman–Kac 类型公式，推广标准自伴随Q是自伴零阶算子的理论。对于紧凑的\(\mathscr {M}\)，我们将此公式与 Berezin 积分相结合，以推导出形式的算子迹的 Feynman-Kac 类型公式

其中\(V,\widetilde{V}\)是零阶，P是阶\(\le 1\)。然后使用这些公式来获得紧凑偶数维黎曼自旋流形的等变陈特征（JLO-cocycle 的微分分级扩展）的低阶项的概率表示，它与循环同调相结合起着至关重要的作用在 Duistermaat-Heckmann 定位公式的上下文中对这种流形的循环空间的作用。