Stochastic equations on projective systems of groups

We consider stochastic equations of the form $X_k={\varphi }_k(X_{k+1})Z_k$, $k \in $\mathbb{N}$$,where$X_k$and$Z_k$ are random variables taking values in a compact group $G_k$, ${\varphi }_k:G_{k+1}\to G_k$ is a continuous homomorphism, and the noise $(Z_k{)}_{k\in \mathbb{N}}$ is a sequence of independent random variables. We take the sequence of homomorphisms and the sequence of noise distributions as given, and investigate what conditions on these objects result in a unique distribution for the “solution” sequence $(X_k{)}_{k\in \mathbb{N}}$ and what conditions permits the existence of a solution sequence that is a function of the noise alone (that is, the solution does not incorporate extra input randomness “at infinity”). Our results extend previous work on stochastic equations on a single group that was originally motivated by Tsirelson’s example of a stochastic differential equation that has a unique solution in law but no strong solutions.