[Paper Review] Some Ergodic Theorems for Random Rotations on Wiener Space

Ali Süleyman Üstünel, Moshe Zakai|ArXiv.org|Oct 16, 2001

Mathematical Dynamics and Fractals6 references3 citations

TL;DR

This paper establishes sufficient conditions for strong mixing and ergodicity of random rotations on Wiener space, using Malliavin calculus and Skorohod integration. It proves that if the random unitary operators satisfy a vanishing correlation condition in probability, the induced transformations are strongly mixing, generalizing classical results to nonlinear, random settings.

ABSTRACT

In this paper we study ergodicity and mixing property of some measure preserving transformations on the Wiener space (W,H,μ) which are generated by some random unitary operators defined on the Cameron-Martin space H.

Motivation & Objective

• To investigate the ergodic and mixing properties of measure-preserving transformations on Wiener space generated by random unitary operators on the Cameron-Martin space.
• To extend classical ergodicity results for deterministic rotations to the case of random, non-linear transformations.
• To characterize strong mixing using conditions on the asymptotic decorrelation of random isometries.
• To provide sufficient conditions for strong mixing in terms of convergence to zero in probability of inner products involving the random rotation operators.
• To establish generic examples of strongly mixing transformations using adapted processes and chaotic representation properties.

Proposed method

• Utilizes the Malliavin calculus framework, particularly the Skorohod integral and divergence operator, to define and analyze random rotations on Wiener space.
• Applies the second quantization of random unitary operators on the Cameron-Martin space to generate measure-preserving transformations.
• Employs the Ito-Nisio theorem to represent Wiener paths as series in an orthonormal basis of the Cameron-Martin space.
• Introduces a condition on the resolution of identity of the random unitary operator to ensure it defines a cylindrical martingale with chaotic representation property.
• Derives a sufficient condition for strong mixing based on the convergence to zero in probability of the inner product (Qₙh, k)ₕ for all h, k ∈ H.
• Uses Girsanov-type identities and the Ito representation theorem to verify the mixing condition in specific examples.

Experimental results

Research questions

• RQ1 Under what conditions is a random rotation on Wiener space strongly mixing?
• RQ2 How can ergodicity be characterized when the unitary operator is random and non-deterministic?
• RQ3 What role does the chaotic representation property of the resolution of identity play in ensuring mixing?
• RQ4 Can the strong mixing condition be reduced to a probabilistic convergence condition on the inner products of rotated vectors?
• RQ5 What are explicit examples of strongly mixing random rotations on Wiener space?

Key findings

• A sufficient condition for strong mixing of a sequence of random rotations (Tₙ) is that (Qₙh, k)ₕ → 0 in probability for all h, k ∈ H.
• The condition (Qₙh, k)ₕ → 0 in probability is also necessary when the rotation operator is deterministic.
• In Example 1, a time-continuous random rotation defined by an adapted process Rₜ with unitary increments is shown to be strongly mixing if the n-th power of the two-point function Aₛ,ₜⁿ → 0 almost surely.
• In Example 2, a sequence of discrete random rotations defined via sign functions of independent Brownian motions satisfies (Qₙh, k)ₕ → 0 in L², hence the sequence is strongly mixing.
• The paper establishes that the chaotic representation property of the cylindrical martingale associated with the resolution of identity is crucial for deriving mixing conditions.
• The results generalize classical ergodicity results for deterministic rotations by incorporating randomness through the unitary operator R(w), which induces non-linear, non-Gaussian dynamics on Wiener paths.