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Operator decomposable measures and stochastic difference equation

We consider the following convolution equation or equivalently stochastic difference equation $$\lam _k = μ_k*ϕ(\lam _{k-1}), k \in \Z \eqno (1) $$ for a given bi-sequence $(μ_k)$ of probability measures on $\R ^d$ and a linear map $ϕ$ on $\R ^d$. We study the solutions of equation (1) by realizing the process $(μ_k)$ as a measure on $(\R ^d)^\Z$ and rewriting the stochastic difference equation as $\lam = μ*τ(\lam )$-any such measure $\lam$ on $(\R ^d)^\Z$ is known as $τ$-decomposable measure with co-factor $μ$-where $τ$ is a suitable weighted shift operator on $(\R ^d)^\Z$. This enables one to study the solutions of (1) in the settings of $τ$-decomposable measures. A solution $(\lam _k)$ of (1) will be called a fundamental solution if any solution of (1) can be written as $\lam _k*ϕ^k(ρ)$ for some probability measure $ρ$ 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 $τ$-decomposable measures on $(\R ^d)^\Z$ with co-factor $μ$. 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 $\ell _p$-paths.