Feynman–Kac formula for perturbations of order $$\le 1$$, and noncommutative geometry
نویسندگان
چکیده
Abstract Let Q be a differential operator of order $$\le 1$$ ≤ 1 on complex metric vector bundle $$\mathscr {E}\rightarrow \mathscr {M}$$ E → M with connection $$\nabla $$ ∇ over possibly noncompact Riemannian manifold . Under very mild regularity assumptions that guarantee ^{\dagger }\nabla /2+Q$$ † / 2 + Q canonically induces holomorphic semigroup $$\mathrm {e}^{-zH^{\nabla }_{Q}}$$ e - z H in $$\Gamma _{L^2}(\mathscr {M},\mathscr {E})$$ Γ L ( , ) (where z runs through sector which contains $$[0,\infty )$$ [ 0 ∞ ), we prove an explicit Feynman–Kac type formula for {e}^{-tH^{\nabla t , $$t>0$$ > generalizing the standard self-adjoint theory where is zeroth operator. For compact ’s combine this Berezin integration to derive trace form $$\begin{aligned} \mathrm {Tr}\left( \widetilde{V}\int ^t_0\mathrm {e}^{-sH^{\nabla }_{V}}P\mathrm {e}^{-(t-s)H^{\nabla }_{V}}\mathrm {d}s\right) \end{aligned}$$ Tr V ~ ∫ s P d $$V,\widetilde{V}$$ are and P These formulae then used obtain probabilistic representations lower terms equivariant Chern character (a graded extension JLO-cocycle) even-dimensional spin manifold, combination cyclic homology play crucial role context Duistermaat–Heckmann localization loop space such manifold.
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ژورنال
عنوان ژورنال: Stochastics And Partial Differential Equations: Analysis And Computations
سال: 2022
ISSN: ['2194-0401', '2194-041X']
DOI: https://doi.org/10.1007/s40072-022-00269-3