Renormalized Differential Geometry on Path Space: Structural Equation, Curvature

Abstract

AbstractThe theory of integration in infinite dimensions is in some sense the backbone of probability theory. On this backbone the stochastic calculus of variations has given rise to the flesh of differential calculus. Its first step is the construction at each point of the probability space of a Cameron–Martin-like tangent space in which the desired differential calculus can be developed. This construction proceeds along the lines of first-order differential geometry. In this paper we address the following questions: what could be the meaning of “curvature of the probability space”—how and why? How can curvatures be defined and computed? Why could a second-order differential geometry be relevant to stochastic analysis? We try to answer these questions for the probability space associated to the Brownian motion of a compact Riemannian manifold. Why? A basicenergy identity for anticipative stochastic integralswill be obtained as a byproduct of our computation of curvature. How? There are essentially four bottlenecks in the development of differential geometry on Wiener–Riemann manifolds: (i) the difficulty of finding an atlas of local charts such that the changes of charts preserve the class of the Wiener-like measures together with their associated Cameron–Martin-like tangent spaces; (ii) the difficulty of finding cylindrical approximations preserving the natural geometrical objects; (iii) the difficulty of renormalizing the divergent series to which the summation operations of finite dimensional differential geometry give rise in the non intrinsic context of local charts; (iv) the nonavailability of the computational procedures analogous to the local coordinates systems of the classical differential geometry. In the context of path space, the Itô filtration provides a much richer structure than that available in the framework of an abstract Wiener–Riemann manifold. Our work is a systematic attempt to replace the machinery of local charts with amethodology of moving frames. In our context, stochastic parallel transport provides a canonical moving frame on the path space. The concept of a cylindrical approximation has to be reshaped in our new situation into some geometric limit theorems, establishing that the Riemannian geometric objects of the cylindrical approximations induce by a limiting procedure geometric objects on the path space. Those limit theorems are reminiscent of the classical theorems which say that a Stratonovich SDE is the limit of an appropriate sequence of ODE. The canonical coordinate system provided by the moving frame will make it possible to proceed to the needed renormalizations byintrinsicstochastic integrals; in this context the anticipative stochastic integral theory of Nualart and Pardoux will play a decisive role. Finally, the moving frame will provide aneffective algorithm of computation for this differential geometry in infinite dimensions. In our study we encounter a new type of renormalization, thehypoelliptic renormalization, which corresponds to the fact that the bracket of smooth vector fields taking their values in the Cameron–Martin space can get out of this Cameron–Martin space. This hypoelliptic problem induces the nonrenormalizability of some geometrical objects. It leads also to a concept oftangent processesto probability spaces extending that based on the Cameron–Martin Theorem. For tangent processes a formula of integration by parts still holds; furthermore the tangent processes form a Lie algebra under the bracket. On the other hand, tangent processes cannot be stochastically integrated: this operation is well defined only for Cameron–Martin-type vector fields