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Integration over paths in $\mathbb{R}^d$ using the Wiener measure seems to be understood in the context of Abstract Wiener Spaces (AWS). These are defined as triples $(\iota,\mathscr{H},E)$ where $\mathscr{H}$ is a separable Hilbert space, $E$ is a separable Banach space and $\iota : \mathscr{H}\to E$ is a continuous injective linear map with dense image which radonifies the canonical Gaussian cylinder measure of $\mathscr{H}$.

The induced measure $\gamma$ on $E$ is the abstract Wiener measure and it is the measure with which the path integral is performed.

Given this context, path integration in $\mathbb{R}^d$ is done by taking $E$ the space of continuous paths $f : [0,1]\to \mathbb{R}^d$ with $f(0)=0$, $\mathscr{H}$ the space of absolutely continuous paths with square integrable derivative equipped with inner product $$\langle f,g\rangle=\int_0^1 \langle f'(t),g'(t)\rangle_{\mathbb{R}^d}dt$$

and $\iota :\mathscr{H}\to E$ is the inclusion.

Now: $E$ and $\mathscr{H}$ are vector spaces because so is $\mathbb{R}^d$ the space on which the paths take values on. This underlying vector space structure is required to make $E$ and $\mathscr{H}$ respectively Banach and Hilbert spaces.

Suppose now we have a Riemannian manifold $(M,g)$ and we want to integrate over paths in $M$. I see a big problem here:

The space of continous paths $f : [0,1]\to M$ with one endpoint fixed at some chosen point $p\in M$, i.e., with $f(0)=p$, does not have a vector space structure.

After all, what would $f(t)+g(t)$ even mean if there's no addition in $M$?

In that case the space of continuous paths doesn't seem to be a Banach space anymore. This seems to break the whole AWS construction.

So the question here is: how can the AWS construction be applied to construct path integrations and Wiener measures on Riemannian manifolds? References for full details are highly appreciated.

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    $\begingroup$ Have you looked at the books on this topic? Search for "Brownian motion on Riemannian manifolds", and/or look for the names Ito, McKean. $\endgroup$
    – Lee Mosher
    Jun 17, 2019 at 18:25
  • $\begingroup$ Thanks for the suggestions. I'm currently using mostly the paper "Abstract Wiener Spaces" by Leonard Gross and an appendix to the book "Quantum Physics: a Functional Integral Point of View" by Glimm and Jaffe. $\endgroup$
    – Gold
    Jun 18, 2019 at 3:08

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I've been through the same difficulties as you, until I realized that the common wisdom of "Wiener measure through AWS" is true only in $\mathbb R^n$, as you suspect. The construction on manifolds is vastly diferent and, in fact, it has nothing to do woth AWS (which I even consider misleading). A full construction can be found in "Wiener Measures on Riemannian Manifolds and the Feynman-Kac Formula" by C. Bär and F. Pfäffle.

The rough construction goes like this: let $M$ be your manifold, fix some $x \in M$ (it will play the role of $0 \in \mathbb R^n$) and denote by $w$ the Wiener measure (to be constructed; it will depend on $x$). Let the heat kernel of $M$ be $h : (0,\infty) \times M \times M \to (0, \infty)$.

1.) Use the Kolmogorov extension theorem to construct a (positive, finite, Borel) measure $w$ on $M^{[0,1]} = \prod _{t \in [0,1]} M$ (visualize this as the space of all trajectories $: [0,1] \to M$ in a purely set-theoretic sense, i.e. regardless of whether they are continuous, measurable or otherwise), that has prescribed pushforwards under the natural projections: if $I = \{0, t_1, \dots, t_{m-1}, 1\}$ and $p_I : M^{[0,1]} \to M^m$ is the natural projection $p_i (c) = (c(0), \dots, c(t_{m-1}))$ (i.e. a "discretization of $c$"), then for any integrable $f : M^m \to \mathbb C$ it is true that

$$\tag{*} \int _{M^{[0,1]}} f \circ p_I \ \mathrm d w = \int _M h(1 - t_{m-1}, x, x_{m-1}) \dots \int _M h(t_1 - 0, x_1, x_0) \ f(x_0, \dots, x_{m-1}) \ \mathrm d x_0 \dots \mathrm d x_{m-1} \ .$$

The function $f \circ p_I$ is called "cylindrical". Notice that $w$ is concentrated on set of trajectories with $c(1) = x$ (fixed endpoint). If you reverse the order of the heat kernels (i.e. putting the innermost in the outermost position, and vice-versa), then $w$ will be concentrated on the trajectories with $c(0) = x$, i.e. fixed starting point. It's just a matter of convention.

Notice also that $w$ is not necessarily a probability. It is true, though, that its mass is $\le 1$. (When $w$ is a probability, $M$ is said to be stochastically complete. There are known sufficient conditions for stochastic completeness, but not necessary and sufficient ones.)

2.) Assume $M$ compact. Show that the Kolmogorov continuity condition is satisfied (check the paper for what this means). Deduce that $w$ is in fact concentrated on the space $\mathcal H$ of Hölder-continuous curves of Hölder exponent $< \frac 1 2$.

In particular, you may extend $w$ (trivially, by $0$) to the space $\mathcal C$ of all continuous curves in $M$. Endowing this space with the distance $D(c, \gamma) = \max _{t \in [0,1]} d(c(t), \gamma (t))$ (with $d$ being the intrinsic distance on $M$), one can show that besides the properties enumerated above, $w$ is also compact-regular (essentially because $\mathcal C$ is Polish and $w$ positive, Borel and finite). It follows that the restriction of $w$ to $\mathcal H$ is also compact-regular. (The only advantage of $\mathcal C$ over $\mathcal H$ is that the former is complete under $D$, while the latter is not, allowing the use of some powerful theorems from descriptive set theory.)

From now on assume $M$ non-compact.

3.) Let $U \subseteq M$ be a regular domain, i.e. a relatively compact open subset with smooth boundary. Using the "doubling trick" (see the paper), and point 2.) show that $U$ (with the induced Riemannian manifold structure) admits a Wiener measure $w_U$ with all the properties listed above.

4.) Consider an exhaustion $M = \bigcup _{k \ge 0} U_k$ with regular domains (such a thing is known to exist). Let $w_k$ be the Wiener measure of $U_k$. Define the Wiener measure $w$ of $M$ to be the limit of $w_k$. More precisely, if $i_k : U_k \hookrightarrow M$ is the natural embedding, then consider the pushforward $(i_k)_* w_k$ on $\mathcal H$ and put $w = \lim_k (i_k)_* w_k$. It is shown that this limit exists and that its pushforwards under the projections $p_I$ (which descend naturally to $\mathcal H$) satisfy formula $(*)$.

Almost all the details are given in the cited article (notice that its authors also treat, in parallel, the case when $M$ has boundary - which makes the article a bit longer than needed). To my knowledge, this poorly known work is the only one constructing rigorously and in full the Wiener measure on arbitrary Riemannian manifolds, starting from geometrical "first principles".

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