On Metrics for Tangent Processes on the Path Space

Abstract

Tangent processes, which are semimartingales with antisymmetric diffusion coefficient, were introduced in the framework of geometry on the path space by Cruzeiro and Malliavin ([3]). They correspond to an extension of the usual Cameron-Martin tangent space in Malliavin calculus, an extension which is in fact necessary in the non-flat situation. In this paper we discuss the possibility of introducing a metric or a Finsler structure on the space of tangent processes. We prove that the Levi—Civita and Cartan connections associated to the natural candidates to the Finsler geometry are not well defined.