Is there a generalization of Brownian motion to general metric spaces (which should probably be length spaces)?


This should be a process satisfying
$$d(B\_t, B\_s) \sim \mathcal{N}(0, t-s)$$
and such that $d(B\_t, B\_s)$ and $d(B\_s, B\_r)$ are independent; however, this will be not enough in general (this not even defines Brownian motion on $\mathbb{R}^n$).


**\Edit:** 1) It was pointed out by Did in the comments that $d(B\_t, B\_s)$ is not normal distributed even in $\mathbb{R}^n$. So this requirement should be different.


2) I wanted to know, if it is possible to define Brownian motion on general metric spaces and if there is some widely accepted definition. Benoît Kloeckner suggested below that one defines the process $B^\varepsilon\_{\tau\_j}$ for some subdivision $\tau$ of $[0, \infty)$, by jumping uniformly in a ball of radius $\varepsilon$. Then take the limit $\varepsilon \longrightarrow 0$ by simultaneously refining the subdivision in an appropriate way.


To me, this seems reasonable and probably is the answer that I was looking for. However, there is no answer, so I cannot accept.


However, he claimed that one should be more precise about the type of metric space. To me, this sounded like there may be other definitions that give different results. It would be interesting to learn about this.



Brownian motion on Metric spaces
I place my comment as answer, as it seems to at least partly satisfy the OP.


If you have a metric *measure* space, you can define a random walk by jumping uniformly in a ball of radius $\varepsilon$, then make $\varepsilon\to0$ while rescaling time to get a continuous stochastic process. But then you certainly need hypotheses to have convergence to a nice analogue of Brownian motion.


To go further, you should tell to what kind of spaces you want to generalize Brownian motion, and which properties are more important to you.


