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.

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    $\begingroup$ 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. $\endgroup$ – Anton Petrunin Dec 4 '18 at 23:46
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    $\begingroup$ @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. $\endgroup$ – MKO Dec 5 '18 at 0:25
  • $\begingroup$ I mean, will we see something we did not see in the RIemannian world? $\endgroup$ – Anton Petrunin Dec 5 '18 at 2:01

Yes, there is a natural Brownian motion on an Alexandrov space.

In the following paper:

Kuwae, Kazuhiro; Machigashira, Yoshiroh; Shioya, Takashi, Sobolev spaces, Laplacian, and heat kernel on Alexandrov spaces, Math. Z. 238, No. 2, 269-316 (2001). ZBL1001.53017.

the authors study the Laplacian on an arbitrary Alexandrov space, and show that it induces a continuous heat kernel. So the Brownian motion should simply be a continuous Markov process whose generator is the Laplacian and whose transition density is given by the heat kernel.

In fact, the authors show that the Laplacian induces a strongly local regular Dirichlet form, and the classical result about such a Dirichlet form is that it is canonically associated with a continuous Markov process having various nice properties (the results of this paper actually imply that it is a Feller process). See

Fukushima, Masatoshi; Oshima, Yoichi; Takeda, Masayoshi, Dirichlet forms and symmetric Markov processes., de Gruyter Studies in Mathematics 19. Berlin: Walter de Gruyter (ISBN 978-3-11-021808-4/hbk; 978-3-11-021809-1/ebook). x, 489 p. (2011). ZBL1227.31001.


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