Wim Vervaat, On a Stochastic Difference Equation and a Representation of Non-Negative Infinitely Divisible Random Variables, Advances in Applied Probability, Vol. 11, No. 4 (Dec., 1979), pp. 750-783 ...

Also, the solutions to linear difference equations over the circle group can be simply described in a fashion which an example shows cannot be extended to all compact abelian groups.