Solutions to complex smoothing equations

We consider smoothing equations of the form $$X ~\stackrel{\mathrm{law}}{=}~ \sum_{j \geq 1} T_j X_j + C$$ where $(C,T_1,T_2,\ldots)$ is a given sequence of random variables and $X_1,X_2,\ldots$ are independent copies of $X$ and independent of the sequence $(C,T_1,T_2,\ldots)$. The focus is on complex smoothing equations, i.e., the case where the random variables $X, C,T_1,T_2,\ldots$ are complex-valued, but also more general multivariate smoothing equations are considered, in which the $T_j$ are similarity matrices. Under mild assumptions on $(C,T_1,T_2,\ldots)$, we describe the laws of all random variables $X$ solving the above smoothing equation. These are the distributions of randomly shifted and stopped L\'evy processes satisfying a certain invariance property called $(U,\alpha)$-stability, which is related to operator (semi)stability. The results are applied to various examples from applied probability and statistical physics.