Itô maps and analysis on path spaces

Abstract

We consider versions of Malliavin calculus on path spaces of compact manifolds with diffusion measures, defining Gross-Sobolev spaces of differentiable functions and proving their intertwining with solution maps, I, of certain stochastic differential equations. This is shown to shed light on fundamental uniqueness questions for this calculus including uniqueness of the closed derivative operator d and Markov uniqueness of the associated Dirichlet form. A continuity result for the divergence operator by Kree and Kree is extended to this situation. The regularity of conditional expectations of smooth functionals of classical Wiener space, given I, is considered and shown to have strong implications for these questions. A major role is played by the (possibly sub-Riemannian) connections induced by stochastic differential equations: Damped Markovian connections are used for the covariant derivatives.

An L2 theory for differential forms on path spaces I

“…In Section 9D we essentially show that D 2,1 H -two-forms are in the domain of the adjoint of d 1 * , extending the result for one-forms proved in [35]. T ξ t -space derivative of ξ t .…”

“…This helps explain the "cancellation" of the bracket occurring with our exterior derivative, and fits in with the result of Cruzeiro and Fang [16], concerning the vanishing of the divergence of such torsions. The damped Markovian connection, introduced by Cruzeiro and Fang [16], plays an important role here, as it did in [35]. As in [35] we introduce it by giving a C id ([0, T ]; O(n))-bundle structure to H. This is done in Section 9.…”

“…The damped Markovian connection, introduced by Cruzeiro and Fang [16], plays an important role here, as it did in [35]. As in [35] we introduce it by giving a C id ([0, T ]; O(n))-bundle structure to H. This is done in Section 9. Here we also relate the divergence of our H -two-vector fields to the adjoint of the damped Markovian covariant derivative in a non-anticipating situation, Corollary 9.7: For suitable non-anticipating U , V ,…”

“…Set H = H 1 . Since these are only almost sure defined it is not strictly speaking correct to consider them as bundles over C x 0 M though some vector bundle structure is given to H in [35] see also Section 9 below. The space of L 2 sections of H q and H q * are denoted by L 2 Γ H q and L 2 Γ H q * .…”

“…Differentiation of functions. For scalar analysis in our context and with this notation, we refer to [35] or for the basic facts to [30]. As emphasised in [35] it is necessary to fix an initial domain, Dom(d H ) ⊂ L 2 (C x 0 M; R) for the H -derivative operator d H .…”