Feynman–Kac formula for perturbations of order ≤1, and noncommutative geometry
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...
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Published in | Stochastic partial differential equations : analysis and computations Vol. 11; no. 4; pp. 1519 - 1552 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
New York
Springer US
01.12.2023
Springer Nature B.V |
Subjects | |
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 2194-0401 2194-041X |
DOI: | 10.1007/s40072-022-00269-3 |