# Independent increments for the Brownian motion on a Riemannian manifold

MathOverflow Asked by Alex M. on October 26, 2020

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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