Stochastic equations on projective systems of groups

Stochastic equations on projective systems of groups

Report Number
803
Authors
Steven N. Evans and Tatyana Gordeeva
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.

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