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???!!!)


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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