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?)

0 Asked on November 3, 2021 by marco-farinati

1 Asked on November 3, 2021 by rodrigo-dias

fa functional analysis fredholm operators index theory kt k theory and homology vector bundles

1 Asked on November 3, 2021 by s-t-stanly

ac commutative algebra ag algebraic geometry dedekind domains nonnoetherian valuation rings

0 Asked on November 3, 2021 by fisura-filozofica

0 Asked on November 3, 2021 by seva

0 Asked on November 3, 2021 by omar-antoln-camarena

at algebraic topology classifying spaces homotopy theory semigroups and monoids

0 Asked on November 3, 2021

analytic number theory co combinatorics generating functions

0 Asked on November 3, 2021

ag algebraic geometry geometric invariant theory orbifolds reference request reflection groups

1 Asked on November 3, 2021

limits and convergence measure theory stochastic calculus stochastic processes

0 Asked on November 3, 2021 by milo-moses

analytic continuation analytic number theory nt number theory

0 Asked on November 3, 2021

co combinatorics combinatorial optimization discrete geometry mg metric geometry permutations

0 Asked on November 3, 2021

0 Asked on November 3, 2021

differential operators elliptic pde heat equation linear pde mp mathematical physics

0 Asked on November 3, 2021

3 Asked on November 3, 2021 by tanya-vladi

fourier analysis fourier transform pr probability real analysis

9 Asked on November 3, 2021 by tom-leinster

0 Asked on November 3, 2021 by yemon-choi

2 Asked on November 3, 2021 by user175348

Get help from others!

Recent Questions

Recent Answers

- haakon.io on Why fry rice before boiling?
- Joshua Engel on Why fry rice before boiling?
- Jon Church on Why fry rice before boiling?
- Peter Machado on Why fry rice before boiling?
- Lex on Does Google Analytics track 404 page responses as valid page views?

© 2022 AnswerBun.com. All rights reserved. Sites we Love: PCI Database, MenuIva, UKBizDB, Menu Kuliner, Sharing RPP, SolveDir