riemannian geometry – Does a random walk on a surface visit uniformly?

Let $$S$$ be a smooth compact closed surface embedded in $$mathbb{R}^3$$ of genus $$g$$.
Starting from a point $$p$$, define a random walk as taking discrete steps
in a uniformly random direction,
each step a geodesic segment of the same length $$delta$$.
Assume $$delta$$ is less than the injectivity radius and small with respect to the intrinsic diameter of $$S$$.

Q. Is the set of footprints of the random walk evenly distributed on $$S$$,
in the limit? By evenly distributed I mean the density of points per unit area of surface
is the same everywhere on $$S$$.

This is likely known, but I’m not finding it in the literature on random walks
on manifolds. I’m especially interested in genus $$0$$.
Thanks!