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In am not a probabilist, but I must do some stochastic-flavoured work on a connected Riemannian manifold $M$. A nice thing about the Brownian motion on $\mathbb R^n$ is that we may talk about its increments, and about them being independent. One consequence of this is that, if $\mathcal C$ is the space of continuous curves $c : [0,1] \to \mathbb R^n$, endowed with the Wiener measure $w$, and if $F : \mathbb R^n \to \mathbb C$ is some function, then

$$\int _{\mathcal C} F(c(s') - c(s)) \ F(c(t') - c(t)) \ \mathrm d w (c) = \int _{\mathcal C} F(c(s') - c(s)) \ \mathrm d w (c) \ \int _{\mathcal C} F(c(t') - c(t)) \ \mathrm d w (c)$$

for any numbers $0 \le s < s' \le t < t' \le 1$. The core ingredients used here are the invariance of the Euclidean heat kernel and of the Lebesgue measure under translations, and the stochastic completeness of $\mathbb R^n$ (the heat semigroup is Markovian).

Is there any substitute for the above formula on a Riemannian manifold considered with its heat kernel?

I need to "decouple" a product like the one in the left hand side of the above equality (with the function $F$ now defined on $M \times M$), and I do not know how to do it, and even whether it is possible to do it in general (it might be necessary to restrict the class of manifolds that I am working on). Or the equality given above could become true only modulo some "small" terms, I don't know.

I know how to do it on Riemannian homogeneous spaces (because I have a notion of invariance under translations), and I am also aware of Erik Jørgensen's "The Central Limit Problem for Geodesic Random Walks". Since I am not a probabilist, though, it is difficult for me to understand whether chapter 3 of this work is relevant to my question. It seems to me that "invariance" should somehow be understood as invariance under parallel transport (or under isometries?), but since Jørgensen requires this invariance to happen along any piecewise-smooth curve, I believe that this imposes severe restrictions on the manifolds that admit it (are they significantly more that just Riemannian homogeneous spaces?)

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  • $\begingroup$ I believe $c(s')-c(s)$ makes no sense for a general manifold $M$. If $M$ is a Lie group (or perhaps a symmetric space), however, then you are good to go. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 2, 2020 at 8:30
  • $\begingroup$ @MateuszKwaśnicki: I know that, this is why I am asking about a substitute. Yes, as I have said, I know how to do it on Riemannian homogeneous spaces. $\endgroup$
    – Alex M.
    Commented Oct 2, 2020 at 8:31
  • $\begingroup$ Ah, sorry, I thought "homogeneous space" means something else. $\endgroup$
    – Mateusz Kwaśnicki
    Commented Oct 2, 2020 at 8:32
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    $\begingroup$ @Alex Well, one generalisation is that harmonic functions (the analogue of the coordinate functions) are martingales (the analogue of having independent increments), but that may not be terribly useful for you... $\endgroup$
    – Martin Hairer
    Commented Oct 2, 2020 at 18:28
  • $\begingroup$ @MartinHairer: Indeed, I am looking for a concept not only of theoretical value, but also of operational one, allowing me to juggle with formulae. So far, Riemannian homogeneous spaces fit the bill. Surprinsingly, though, I am unable to say anything about closed manifolds, even though these are among the nicest Riemannian objects. Maybe the plain Riemannian structure is too amorphous, and one really needs some supplementary algebraic structure allowing one to relate points to each other? $\endgroup$
    – Alex M.
    Commented Oct 2, 2020 at 18:57

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