A Cameron-Martin type quasi-invariance theorem for pinned Brownian motion on a compact Riemannian manifold

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.