Stochastic equations on projective systems of groups

TL;DR

This paper studies stochastic equations on projective systems of groups, analyzing conditions for unique solutions and solutions determined solely by noise, extending prior work on stochastic equations on single groups.

Contribution

It extends previous results by characterizing conditions for uniqueness and noise-determined solutions in stochastic equations on projective systems of groups.

Findings

Conditions for unique distribution of solutions.

Criteria for solutions depending only on noise.

Extension of Tsirelson's example to group systems.

Abstract

We consider stochastic equations of the form $X_k = \phi_k(X_{k+1}) Z_k$, $k \in \mathbb{N}$, where $X_k$ and $Z_k$ are random variables taking values in a compact group $G_k$, $\phi_k: G_{k+1} \to G_k$ 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.