Right now I am learning about analysis of stochastic processes and the Malliavin calculus. It seems though, that most of the theory works for Brownian motion in $\mathbb{R}^n$, and it seems non-obvious to generalize these things to Brownian motion on Riemannian manifolds.

Let $M$ be a Riemannian manifold and let $W \subseteq C([0, T], M)$ be some manifolds of continuous paths, say those with one or two fixed endpoints. Now I would like to define a vector bundle $\mathcal{H}$ of $H^1$ paths over $W$ (this bundle should somehow play the role of the Cameron-Martin space). At each point $\omega \in W$, $\mathcal{H}_\omega$ should be a subspace of $T_\omega W$, the "space of vector fields along $\omega$ of Sobolev regularity $H^1$". However, this does not make sense, because these paths should be sections of the vector bundle $\omega^*TM$ over $[0, T]$, but for a general continuous path, this is not a smooth vector bundle, so it is not clear what the regularity of a section higher than continuous would even mean.

Of course, one would also like to equip it with the scalar product $$( X, Y ) = \int_0^T \left\langle \frac{\nabla}{\mathrm{d} t} X, \frac{\nabla}{\mathrm{d} t} Y \right\rangle \mathrm{d} t$$ which a priori also doesn't make much sense (how does one differentiate along a not differentiable path??).

So one question is: How to construct a vector bundle of $H^1$ vector fields along paths over $W$?

What would you want to do with it? For example, you would like to proof a statement similar to the following: If $W = W_x$ is the space of paths starting at a fixed point $x \in M$ equipped with the Wiener measure, the section $\omega \mapsto \int \dot{\omega}$ where I denote $\bigl(\int \dot{\omega}\bigr)(t):=\int_0^t \dot{\omega}(s) \mathrm{d}s$ is an $L^2$ section of $\mathcal{H}$, i.e. an element of $L^2(W, \mathcal{H})$. Hence for any $X \in L^2(W, \mathcal{H})$, the pointwise scalar product $(X, \int \dot{\omega})$ exists as a measurable function on $W$, which is just the Ito integral $$ \bigl(X, \int\dot{\omega}\bigr) = \int_0^T \left\langle\frac{\nabla}{\mathrm{d}t}X(t), \mathrm{d}B_t\right\rangle.$$ These are just very rough ideas and certainly wrong in several ways, but problably someone can formulate the correct version of the claims I made.

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    $\begingroup$ (I can't help much, but love to get a refresher on this, after 10+y. Book recommendations? My fav books are: E.P.Hsu, Stoch. An. on Mfs. (2001), Shigekawa, Stoch. An. (2004) and Malliavin/Thalmaier (2005) Ch. 1.) You need stochastic parallel transport. $\endgroup$ – Martin Gisser Jan 22 '15 at 23:56
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    $\begingroup$ 1.: You need stochastic parallel transport. DARN INTERFACE --- 2.: Definition of vector fields and differentiable funcions go hand in hand here, starting with finite-dim. cylinder fcts. and then doing functional analysis. --- I guess you can forget extending fin.-dim. vector bundle stuff, even standard sheaf thry is limited. --- Your point of entry would be Hsu, §8.4 $\endgroup$ – Martin Gisser Jan 23 '15 at 0:10
  • $\begingroup$ So you believe that the claims I made cannot be made rigorous, i.e. in particular, there exists no such thing as an $H^1$ vector bundle over a $C^0$ path space? What do you mean with "standard sheaf theory is limited"? $\endgroup$ – Matthias Ludewig Jan 23 '15 at 8:35
  • $\begingroup$ The construction of the Cameron-Martin space in a manifold setting was rigorously carried out by P. Malliavin, A.B, Cruzeiro and B. Driver in the 1990's, when they developed the notion of adapted vector fields and tangent processes. The basic idea is to use the stochastic parallel transport. A good reference is the book "Stochastic analysis" by Malliavin himself. The last part of the book contains the basic definitions and the relevant pointers in the literature. $\endgroup$ – Fabrice Baudoin Jan 25 '15 at 20:45

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