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Maybe someone will be able to suggest something about the following quite simple problem:

1D problem on axis "X". Particle moves only along "X" axis and starts its motion from X=0. However, when "X<0" particle disappears. Particle is influenced by some kind of force in such way that we know only the density probability function of particle velocity F[v]. In the case, when the average velocity (can be found from F[v]) is negative, we can conclude that any particle after some time eventually will disappear, since the probability to go to negative "X" is bigger.

The question: if we have infinite number of such particles (and as a result infinite number of trajectories that ends on X<0), what is distribution function for these particles on the axis "X"?

Thank you very much for any suggestions and relevant to this problem sources.