Coagulation Processes with Gibbsian Time Evolution

Abstract

We prove that a stochastic process of pure coagulation has at any time t ≥ 0 a timedependent 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.