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Let $(M,g)$ be a Riemannian manifold, and $E \to M$ be a vector bundle endowed with a connection $\nabla$. If $c:[0,1] \to M$ is a continuous curve, and if $\Delta = \{t_1, \dots, t_m\} \subset [0,1]$, then for small enough $\| \Delta \| = \sup_i (t_{i+1} - t_i)$ the points $c(t_i)$ and $c(t_{i+1})$ may be joined by a unique minimizing geodesic, for all $i$. One thus gets a new piecewise geodesic curve $c_\Delta$. If $v \in E_{c(0)}$ then one may consider its parallel transport $v_\Delta \in E_{c(1)}$ along $c_\Delta$. Denote this parallel transport operator by $U_\Delta (c) : E_{c(0)} \to E_{c(1)}$.

I have seen stated ([1], [2]) that $U_\Delta$ has a limit in measure (the Wiener measure, that is), called the stochastic parallel transport. Nevertheless, I have not seen a rigorous statement of this.

In what space do the objects $U_\Delta$ live? What is the topology on it, in which we consider the limit $\lim _{\|\Delta\| \to 0} U_\Delta$?

I know what convergence in measure means for functions, but the objects $U_\Delta$ are not functions, so I would like to see a very precise statement of what that convergence means. Precise bibliographic references would count as an answer.

[1] "Stochastic Parallel Displacement" - K. Itō

[2] "Stochastic Calculus" - K. Itō

Both texts by Itō are short and general presentations of the subject, extremely light on details, unusable as references.

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  • $\begingroup$ The $U^\Delta$ live in (a slight variation of) what is called the orthonormal frame bundle. This is a manifold, hence a Polish space, and convergence in distribution is well-defined. A classic reference for these matters is the book of E. P. Hsu, Stochastic Analysis on Manifolds, although he does not discuss geodesic approximations, as far as I recall. $\endgroup$
    – Pierre PC
    Commented Jun 23, 2022 at 8:08
  • $\begingroup$ @PierrePC: I don't think so. Notice that $U_\Delta$ depends on the curve $c$, not on points of the manifold. What you can say is that $U_\Delta (c)$ lives in $E_{c(1)}$ (nothing to do with the frame bundle associated to $E$). $\endgroup$
    – Alex M.
    Commented Jun 23, 2022 at 8:30
  • $\begingroup$ It may be, but then I don't understand your notations. Is $U^\Delta(c)$ not a linear map, rather than a point in $E_{c(1)}$? I think this means (up to identifying $E_{c(0)}$ with some $\mathbb R^k$) that $U^\Delta(c)$ is a point in the frame bundle associated to $E$. Then $U^\Delta(\text{some random curve})$ is a random element of the frame bundle, and if the limit exists it will be as an element of the frame bundle. My apologies if this is not what you mean, hopefully I can understand what you mean with more context. $\endgroup$
    – Pierre PC
    Commented Jun 23, 2022 at 11:48
  • $\begingroup$ Also $t\mapsto U^\Delta(c^t)$ is a curve in the frame bundle, for $c^t:u\mapsto c(tu)$, so you can see it as an element of the path space of the frame bundle (say with the uniform topology) and it is again naturally Polish. $\endgroup$
    – Pierre PC
    Commented Jun 23, 2022 at 11:51
  • $\begingroup$ @PierrePC: Indeed, $U_\Delta (c)$ is a linear map, my above comment was rushed. If one fixes some $v \in E_{c(0)}$ (and $c(0)=x_0$ is fixed), then $c \mapsto U_\Delta (c) (v)$ is a section of the pull-back bundle $p^* E$ over the space $C$ of continuous curves with $c(0) = x_0$, where $p : C \to M$ is given by $p(c) = c(1)$. I would like to know whether this is the point of view adopted by the stochastic community, or they view the problem differently. $\endgroup$
    – Alex M.
    Commented Jun 23, 2022 at 11:54

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