The results in Driver [13] for quasi-invariance of Wiener measure on the path space of a compact Riemannian manifold (M) are extended to the case of pinned Wiener measure. To be more explicit, let
h
:
[
0
,
1
]
→
T
0
M
h:[0,1] \to {T_0}M
be a
C
1
{C^1}
function where M is a compact Riemannian manifold,
o
∈
M
o \in M
is a base point, and
T
o
M
{T_o}M
is the tangent space to M at
o
∈
M
o \in M
. Let
W
(
M
)
W(M)
be the space of continuous paths from [0,1] into M,
ν
\nu
be Wiener measure on
W
(
M
)
W(M)
concentrated on paths starting at
o
∈
M
o \in M
, and
H
s
(
ω
)
{H_s}(\omega )
denote the stochastic-parallel translation operator along a path
ω
∈
W
(
M
)
\omega \in W(M)
up to "time" s. (Note:
H
s
(
ω
)
{H_s}(\omega )
is only well defined up to
ν
\nu
-equivalence.) For
ω
∈
W
(
M
)
\omega \in W(M)
let
X
h
(
ω
)
{X^h}(\omega )
denote the vector field along
ω
\omega
given by
X
s
h
(
ω
)
≡
H
s
(
ω
)
h
(
s
)
X_s^h(\omega ) \equiv {H_s}(\omega )h(s)
for each
s
∈
[
0
,
1
]
s \in [0,1]
. One should interpret
X
h
{X^h}
as a vector field on
W
(
M
)
W(M)
. The vector field
X
h
{X^h}
induces a flow
S
h
(
t
,
∙
)
:
W
(
M
)
→
W
(
M
)
{S^h}(t, \bullet ):W(M) \to W(M)
which leaves Wiener measure
(
ν
)
(\nu )
quasi-invariant, see Driver [13]. It is shown in this paper that the same result is valid if
h
(
1
)
=
0
h(1) = 0
and the Wiener measure
(
ν
)
(\nu )
is replaced by a pinned Wiener measure
(
ν
e
)
({\nu _e})
. (The measure
ν
e
{\nu _e}
is proportional to the measure
ν
\nu
conditioned on the set of paths which start at
o
∈
M
o \in M
and end at a fixed end point
e
∈
M
e \in M
.) Also as in [13], one gets an integration by parts formula for the vector-fields
X
h
{X^h}
defined above.