Abstract
The notions of complementability and maximality were introduced in 1974 by Ornstein and Weiss in the context of the automorphisms of a probability space, in 2008 by Brossard and Leuridan in the context of the Brownian filtrations, and in 2017 by Leuridan in the context of the poly-adic filtrations indexed by the non-positive integers. We present here some striking analogies and also some differences existing between these three contexts.
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Notes
- 1.
Actually, Rokhlin’s theory ensures that if \(\mathcal {B}\) is a factor of a Lebesgue space \((Z,\mathcal {Z},\pi ,T)\), then there exists a map f from Z to some Polish space E such that \(\mathcal {B}\) is generated up to the negligible events by the map Φ : x↦(f(Tk(x)))k ∈Z from Z to the product space EZ. Call ν = Φ(π) = π ∘ Φ−1 the image measure of μ by Φ. Then the completion \((E^{\mathbf {Z}},\mathcal {B}(E^{\mathbf {Z}}),\nu )\) is a Lebesgue space, the shift operator S : (yk)k ∈Z↦(yk+1)k ∈Z is an automorphism of EZ, and S ∘ Φ = Φ ∘ T.
Conversely, if \((Y,\mathcal {Y},\nu ,S)\) is a dynamical system and Φ a measurable map from Z to Y such that Φ(π) = ν and S ∘ Φ = Φ ∘ T, then the σ-field \(\Phi ^{-1}(\mathcal {Y})\) is a factor of \((Z,\mathcal {Z},\pi ,T)\).
- 2.
The inclusion \(\mathcal {U}_t = \mathcal {U}_\infty \cap \mathcal {Z}_t\) is immediate. To prove the converse, take \(A \in \mathcal {U}_\infty \cap \mathcal {Z}_t\). Since \(\mathcal {U}_\infty \) and \(\mathcal {Z}_t\) are independent conditionally on \(\mathcal {U}_t\), we get \(\mathbf {P}[A|\mathcal {U}_t] = \mathbf {P}[A|\mathcal {Z}_t] = {\mathbf {1}}_A\) a.s., so \(A \in \mathcal {U}_t \mod \mathbf {P}\).
- 3.
Let Z be a n-dimensional Brownian motion and B be a m-dimensional Brownian motion in \(\mathcal {F}^Z\). Then one can find an \(\mathcal {F}^Z\)-predictable process M taking values in the set of all p × n real matrices whose rows form an orthonormal family, such that \(B = \int _0^\cdot M_s \mathrm {d} Z_s.\) In particular, the m rows of each matrix Ms are independent and lie in a n-dimensional vector space, so m ≤ n.
- 4.
Aperiodicity of T means that π{z ∈ Z : ∃n ≥ 1, Tn(z) = z} = 0. We make this assumption to ensure the existence of generator.
- 5.
Taking logarithms in base 2 is an arbitrary convention which associates one unity of information to any uniform Bernoulli random variable.
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Acknowledgements
I thank A. Coquio, J. Brossard, M. Émery, S. Laurent, J.P. Thouvenot for their useful remarks and for stimulating conversations.
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Leuridan, C. (2019). Complementability and Maximality in Different Contexts: Ergodic Theory, Brownian and Poly-Adic Filtrations. In: Donati-Martin, C., Lejay, A., Rouault, A. (eds) Séminaire de Probabilités L. Lecture Notes in Mathematics(), vol 2252. Springer, Cham. https://doi.org/10.1007/978-3-030-28535-7_6
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