Stochastic Equations on Projective Systems of Groups

DOI10.1007/978-1-4614-6240-8_2zbMATH Open1271.60079arXiv1106.2837OpenAlexW1923234020MaRDI QIDQ5326164

Publication date: 31 July 2013

Published in: Springer Proceedings in Mathematics & Statistics (Search for Journal in Brave)

Abstract: We consider stochastic equations of the form

X_{k} = p h i_{k} (X_{k + 1}) Z_{k}

,

k i n m a t h b b N

, where

X_{k}

and

Z_{k}

are random variables taking values in a compact group

G_{k}

,

p h i_{k} : G_{k + 1} o G_{k}

is a continuous homomorphism, and the noise

(Z_{k})_{k i n m a t h b b 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 i n m a t h b b 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.

Full work available at URL: https://arxiv.org/abs/1106.2837

zbMATH Keywords

extreme point group representation strong solution uniqueness in law toral automorphism Lucas theorem

Mathematics Subject Classification ID

Random operators and equations (aspects of stochastic analysis) (60H25) Probability measures on groups or semigroups, Fourier transforms, factorization (60B15)