# Is there Brownian motion on Alexandrov spaces?

It is well known that there is a notion of Brownian motion on smooth Riemannian manifolds.

I am wondering if there is a more general notion of Brownian motion on finite dimensional Alexandrov spaces with curvature bounded below.

Such spaces are roughly Riemannian “manifolds” with metric and topological singularities. For example convex hypersurfaces are examples of such spaces. In particular boundaries of convex polytopes are.

• The heat flow and harmonic functions can be defined on Alexandrov space, so I do not see a problem to define Brownian motion. Usually people define something for Alexandrov spaces when they expect some new phenomena, but I do not expect anything new for Brownian motion. – Anton Petrunin Dec 4 '18 at 23:46
• @AntonPetrunin: Well, do you think that Brownian motion led to new phenomena on smooth Riemannian manifolds? If yes, why it cannot lead to them for Alexandrov spaces? If no, then probably it might be the case for Alexandrov spaces too. – MKO Dec 5 '18 at 0:25
• I mean, will we see something we did not see in the RIemannian world? – Anton Petrunin Dec 5 '18 at 2:01