Choquet-Deny type functional equations with applications to stochastic models

The ICFE was originally introduced to characterize a probability distribution by some invariant property under a stochastic change (damage) to the original random variable, and it is a generalization of a certain version of the Choquet-Deny convolution equation which occurs in potential theory. The solution of the ICFE is obtained using certain properties of exchangeable random elements or martingales, amongst other things. The solutions to these functional equations provide a unified and elegant approach to characterizations of the exponential, geometric, Pareto, Weibull, stable, Poisson ...