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.

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