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 $\mathbb{R}^n$).