Stochastic Equations on Projective Systems of Groups

We consider stochastic equations of the form Xk = φk(Xk+1)Zk, k ∈ N, where Xk and Zk are random variables taking values in a compact group Gk, φk : Gk+1 → Gk is a continuous homomorphism, and the noise (Zk)k∈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 (Xk)k∈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.