Coagulation processes with Gibbsian time evolution

Abstract

We prove that a stochastic process of pure coagulation has at any time t ≥ 0 a time-dependent Gibbs distribution if and only if the rates ψ(i, j) of single coagulations are of the form ψ(i; j) = if(j) + jf(i), where f is an arbitrary nonnegative function on the set of positive integers. We also obtain a recurrence relation for weights of these Gibbs distributions that allow us to derive the general form of the solution and the explicit solutions in three particular cases of the function f. For the three corresponding models, we study the probability of coagulation into one giant cluster by time t > 0.