Fix a unit vector $v\in\mathbb{R}^3$, a small positive real number $\theta$, and let $g_x,g_y,g_z\in SO(3)$ be rotations around the x,y,z axes by an angle of $\theta$. Let $P_n^\theta$ be the finite distribution of finite random walks on the unit sphere of length $n$ that start at $v$, and at each step we apply one of $g_x,g_y,g_z$, chosen uniformly.
Also assume that $\theta$ isn't a rational multiple of $\pi$ so that we don't get into a weird periodic situation.
Experimenting with a computer, it looks like for $n$ large enough, the distribution $P_n^\theta$ is close to uniform on the unit sphere. Is this true?
Furthermore, experimenting with how many steps this takes, on the range $\theta\in(0.05,0.25)$ it looks a bit like it takes close to $100/\theta$ steps until a nice uniform picture emerges. This function is of course also close to other functions in $\theta$, and it's not clear to me that if such a convergence exists then it should be inverse linear. How many steps does it take for such a random walk on the unit sphere to come close to uniform?