Lets consider the following problem. Random walkers move in a unit sphere of dimension d by jumps with constant length l. Jumps occur uniformly in all direction. A random walker jumps with probability u per time step. If a jump leads outside the unit sphere the random walker is deleted and a randomly chosen random walker inside the unit sphere is copied in order to keep the population constant. After enough time steps a stationary state is reached. Numerically I could show, that the density of random walkers of the stationary state depends on the distance to the center. Now I am curious, if an analytical expression for the density can be derived. So far I have been unsuccessful. I am especially interested in the two-dimension case.
2026-02-22 20:42:11.1771792931
Density distribution of random walkers in a unit sphere with an absorbing boundary
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