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.