Aniello Fedullo, Vitalii A. Gasanenko
We have random number of independent diffusion processes with absorption on boundaries in some region at initial time $t=0$. The initial numbers and positions of processes in region is defined by Poisson random measure. It is required to estimate of number of the unabsorbed processes for the fixed time \~$τ>0$. The Poisson random measure depends on $τ$ and $τ\to\infty$.
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We consider a discrete random walk on a diagonal lattice in two and three dimensions and obtain explicit solutions of absorption probabilities and probabilities of return in several domains. In three dimensions we consider both the cube and the dodecahedron variant. In two dimensions we obtain explicit formula in case of rotated barriers.
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Suppose we have three independent copies of a regular diffusion on $[0,1]$ with absorbing boundaries. Of these diffusions, either at least two are absorbed at the upper boundary or at least two at the lower boundary. In this way, they determine a majority decision between 0 and 1. We show that the strategy that always runs the diffusion whose value is currently between the other two reveals the majority decision whilst minimizing the total time spent running the processes.
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