This is the extension of: Expected square of frog's distance to the centre in the limit of infinite number of jumps: $\lim\limits_{n\to\infty}E[|r_n|^2]$
A frog jumps inside a unit circle (an outside version is also worth studying). The frog chooses a uniform point on the circle and jumps to the middle of the segment connecting its current position and the randomly chosen point.
What is the limiting density of the distance to the origin?
I've tried to write an identity for densities before and after the jump but have not succeeded.