Operator decomposable measures and stochastic difference equation

We consider the following convolution equation or equivalently stochastic difference equation $${\text{lam}}_k={\mu }_k\ast \varphi ({\text{lam}}_{k-1}),k\in \mathbb{Z}\text{eqno}(1)$$ for a given bi-sequence $({\mu }_k)$ of probability measures on ${\mathbb{R}}^d$ and a linear map $\varphi$ on ${\mathbb{R}}^d$. We study the solutions of equation (1) by realizing the process $({\mu }_k)$ as a measure on $({\mathbb{R}}^d{)}^{\mathbb{Z}}$ and rewriting the stochastic difference equation as $\text{lam}=\mu \ast \tau (\text{lam})$-any such measure $\text{lam}$ on $({\mathbb{R}}^d{)}^{\mathbb{Z}}$ is known as $\tau$-decomposable measure with co-factor $\mu$-where $\tau$ is a suitable weighted shift operator on $({\mathbb{R}}^d{)}^{\mathbb{Z}}$. This enables one to study the solutions of (1) in the settings of $\tau$-decomposable measures. A solution $({\text{lam}}_k)$ of (1) will be called a fundamental solution if any solution of (1) can be written as $\lam _k*\phi ^k(\rho )$forsomeprobabilitymeasure$\rho $on$\R ^d$. Motivated by the splitting/factorization theorems for operator decomposable measures, we address the question of existence of fundamental solutions when a solution exists and answer affirmatively via a one-one correspondence between fundamental solutions of (1) and strongly $\tau$-decomposable measures on $({\mathbb{R}}^d{)}^{\mathbb{Z}}$ with co-factor $\mu$. We also prove that fundamental solutions are extremal solutions and vice versa. We provide a necessary and sufficient condition in terms of a logarithmic moment condition for the existence of a (fundamental) solution when the noise process is stationary and when the noise process has independent ${\mathrm{\ell }}_p$-paths.