On the Semi-Classical Brownian Bridge Measure

We prove an integration by parts formula for the probability measure induced by the semi-classical Riemmanian Brownian bridge over a manifold with a pole. 1 Introdcution Let M be a finite dimensional smooth connected complete and stochastically complete Riemannian manifold M whose Riemannian distance is denoted by r. By stochastic completeness we mean that its minimal heat kernel satisfies that ∫ pt(x, y)dy = 1. Denote by C([0, 1];M ) the space of continuous curves: σ : [0, 1] → M , a Banach manifold modelled on the Wiener space. A chart containing a path σ is given by a tubular neighbourhood of σ and the coordinate map is induced from the exponential map given by the Levi-Civita connection on the underlying finite dimensional manifold. For x0, y0 ∈ M we denote by Cx0M and Cx0,y0M , respectively, the based and the pinned space of continuous paths over M : Cx0M = {σ ∈ C([0, 1];M ) : σ(0) = x0}, Cx0,y0M = {σ ∈ C([0, 1];M ) : σ(0) = x0, σ(1) = y0}. The pullback tangent bundle of Cx0M consisting of continuous v : [0, 1] → TM with v(0) = 0 and v(t) ∈ Tσ(t)M where σ ∈ C([0, 1];M ) which for each σ can be identified by parallel translation with continuous paths on Tx0M , the latter is identified with R with a frame u0. To define gradient operators we make a choice of a family of L sub-spaces together with an Hilbert space structure, and so we have a family of continuously embedded L subspacesHσ and the L sub-bundleH := ∪σHσ . Firstly denote by H the Cameron-Martin space over R, H := { h ∈ C([0, 1];R) : h(0) = 0, |h|H1 := (∫ 1 0 |ḣs|ds ) 1 2 <∞ } , with H its subset consisting of h with h(1) = 0. If //·(σ) denotes stochastic parallel translation along a path σ we denote byHσ andH σ the Bismut tangent spaces: Hσ = {//·(σ)h : h ∈ H}, H σ = {//·(σ)h : h ∈ H,h(1) = 0}, AMS Mathematics Subject Classification : 60Dxx, 60 H07, 58J65, 60Bxx