Operator Decomposable Measures and Stochastic Difference Equations

Abstract

We consider the following convolution equation (or equivalently stochastic difference equation) 1 $$\begin{aligned} \lambda _k = \mu _k*\phi (\lambda _{k-1}),\quad k \in {\mathbb Z}\end{aligned}$$ for a given bi-sequence $$(\mu _k)$$ of probability measures on $${\mathbb R}^d$$ and a linear map $$\phi $$ on $${\mathbb R}^d$$ . We study the solutions of Eq. (1) by realizing the process $$(\mu _k)$$ as a measure on $$({\mathbb R}^d)^{\mathbb Z}$$ and rewriting the stochastic difference equation as $$\lambda = \mu *\tau (\lambda )$$ -any such measure $$\lambda $$ 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 $$(\lambda _k)$$ of (1) will be called a fundamental solution if any solution of (1) can be written as $$\lambda _k*\phi ^k(\rho )$$ for some probability measure $$\rho $$ on $${\mathbb R}^d$$ . 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 $$\ell _p$$ -paths.