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