# Brownian motion on Metric spaces

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.

• 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. Without more motivation about what you want from it, it is difficult to give a reasonable answer. – Benoît Kloeckner Jul 3 '13 at 15:15
• Well, I would like a definition that reduces to the case of usual Brownian motion on a Riemannian manifold, if the metric space happens to have this structure. Your definition does the trick, I would say, doesn't it? – Matthias Ludewig Jul 3 '13 at 18:39
• There is something studied under the name "Brownian motion on a fractal" ... but generally this in done in settings far less general than a "metric space" ... proba.jussieu.fr/bulletin/Bmfrac.pdf – Gerald Edgar Jul 3 '13 at 20:35
• A distance distributed by a normal distribution? Usually one is unhappy with negative distances... – Did Jul 3 '13 at 21:00
• For the Riemannian case, see An Introduction to the Analysis of Paths on a Riemannian Manifold by Daniel Stroock, which essentially develops @Benoît's idea. Generalizing this to every metric space is problematic, for example when every ball of radius $\lt1$ is reduced to a point (which might be a reason to ask that you provide more context)... – Did Jul 4 '13 at 9:20

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.

The correct point of view on Brownian motion that generalizes on metric spaces is to see the Brownian motion as the Hunt process in $\mathbb{R}^n$ generated by the Dirichlet form $\int \| \nabla f \|^2 (x) dx$. In a similar way, the Brownian motion on a Riemannian manifold $\mathbb{M}$ is the Hunt process in $\mathbb{R}^n$ generated by the Dirichlet form $\int_{\mathbb{M}} \| \nabla f \|^2 (x) dx$. More generally, once we have a "natural and nice" Dirichlet form on a metric space, one can construct an associated continuous Markov process. You will find some details and relevant references in the paper by Sturm about diffusion processes on metric measure spaces

https://projecteuclid.org/euclid.aop/1022855410

If you use this construction on fractals for instance, you will get the Brownian motion on the fractal, which is a very fun object to study. You can for instance have a look into the short survey by Kumagai

http://www.proba.jussieu.fr/bulletin/Bmfrac.pdf

and the references therein.

I think many people are thinking of this question especially after developments on the subject of metric spaces Lott, Villani, Sturm, Ambrosio, Gigli....!! This will be the first step towards a theory of stochastic analysis on metric spaces. In the case of metric graphs this has been done very carefully starting from the Brownian motion on a star graph introduced by John Walsh (see on Arxiv Brownian motion on metric graphs). The definition on star graphs is based on the existence of a set a vertices. I think nice properties of the space should be needed in order to do this.