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I'm starting to work in random walks and I have two big questions I would like to have suggestions about.

(1) Consider a system of N point particles undergoing Brownian motion (random walks) on a 3D space with equal diffusion coefficients D and an average spacing L. If we have a probability p of these particles merging when they are found at a distance [d1,d2], how many particles will we have after a time t?

(2) Consider a system of two point particles undergoing random walks BUT confined to a sphere of radius a each, which intersect in some section. How can we estimate their first passage time - that is, the time at which they would be found at a distance [d1,d2]?

Since now thanks for any ideas or suggestions or for pointing me to articles!! The guys in physics stackexchange were all like "this is homework" (wtf???!!!)

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The behavior of this model can be inferred from the corresponding continuous-time discrete-space model analyzed in [1]-[4] below.

[1] Van den Berg, J., and Harry Kesten. "Asymptotic density in a coalescing random walk model." Annals of Probability (2000): 303-352.

[2] Van den Berg, J., and Harry Kesten. "Randomly coalescing random walk in dimension≥ 3." In In and out of equilibrium, pp. 1-45. Birkhäuser, Boston, MA, 2002.

[3] Csáki, E., Révész, P. and Shi, Z., 2004. Large void zones and occupation times for coalescing random walks. Stochastic processes and their applications, 111(1), pp.97-118.

[4] Kesten, H., 2000. Coalescing and annihilating random walk with ‘action at a distance’. Journal d’Analyse Mathématique, 80(1), pp.183-256.

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