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!

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!

Update (6JUn2021). The answer to Q is Yes, going back 38 years to Toshikazu Sunada, as recounted in @RW's answer.