Feynman–Kac formula for perturbations of order ≤1, and noncommutative geometry

• Published in: Stochastic partial differential equations : analysis and computations Vol. 11; no. 4; pp. 1519 - 1552
• Main Authors: Boldt, Sebastian, Güneysu, Batu
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.12.2023; Springer Nature B.V
• Subjects: Computational Mathematics and Numerical Analysis; Computational Science and Engineering; Differential equations; Differential geometry; Homology; Manifolds (mathematics); Mathematics; Mathematics and Statistics; Numerical Analysis; Operators (mathematics); Partial Differential Equations; Perturbation; Probability Theory and Stochastic Processes; Riemann manifold; Statistical Theory and Methods; 58J05; Secondary 47D03; Feynman-Kac formula; holomorphic semigroups; Primary 58J65; noncommutative geometry
• ISSN: 2194-0401; 2194-041X
• DOI: 10.1007/s40072-022-00269-3

Let Q be a differential operator of order ≤ 1 on a complex metric vector bundle E → M with metric connection ∇ over a possibly noncompact Riemannian manifold M . Under very mild regularity assumptions on Q that guarantee that ∇ † ∇ / 2 + Q canonically induces a holomorphic semigroup e - z H Q ∇ in Γ L 2 ( M , E ) (where z runs through a complex sector which contains [ 0 , ∞ ) ), we prove an explicit Feynman–Kac type formula for e - t H Q ∇ , t > 0 , generalizing the standard self-adjoint theory where Q is a self-adjoint zeroth order operator. For compact M ’s we combine this formula with Berezin integration to derive a Feynman–Kac type formula for an operator trace of the form Tr V ~ ∫ 0 t e - s H V ∇ P e - ( t - s ) H V ∇ d s , where V , V ~ are of zeroth order and P is of order ≤ 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.