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Erschienen in:
2013 | OriginalPaper | Buchkapitel
Stochastic Equations on Projective Systems of Groups
verfasst von : Steven N. Evans, Tatyana Gordeeva
Erschienen in: Advances in Superprocesses and Nonlinear PDEs
Verlag: Springer US
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Abstract
We consider stochastic equations of the form X
k
= ϕ
k
(X
k + 1)Z
k
, k ∈ ℕ, where X
k
and Z
k
are random variables taking values in a compact group G
k
, ϕ
k
:G
k + 1 → 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 permit the existence of a solution sequence that is a function of the noise alone (i.e., 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.