Integral of Differential Forms along the Path of Diffusion Processes

Abstract

Let (Xt, S t, PX) be a Brownian motion on a Riemannian manifold M and a be a differential 1-form on M. In this paper we will be concerned with the integral of a along Xt. This integral is a stochastic version of the ordinary integral of the form a along a smooth curve on M and is defined by using the symmetric integral. We denote by A(t\a), tl>Q, the integrals of a along Xt. The one-parameter family A= {A(t a); £^>0} of random variables then defines a continuous additive functional of (Xt, 2^, P,) . In Section 3 we will show that A(t; a), tl>0, can be decomposed into a sum of a local martingale and a bounded variation process which is expressed by the divergence of a. The structure of the local martingale part will be analyzed by using the lifted diffusion (rt, 3t, Pr) on O(M) of (Xt, S t, PX) through the Riemannian connection where O(M) is the bundle of orthonormal frames. Next in Section 4, using some results in Section 3 we will give a representation theorem for continuous square integrable martingale additive functionals of (Xt9 31, PX) which was obtained, in some special cases, by a number of authors (cf. [11], [14], [15], [16], [17]). As an application of Theorem 3. 1, we discuss in Section 5 the Cameron-Martin formula. An approximation theorem similar to Nakao-Yamato [12] also holds in our case. Using this we will formulate and prove a stochastic version of Stokes' theorem. M. Yor [18] recently discussed a closely related subject independently in the case that M= C.