[Paper Review] On Spectra of Noises associated with Harris flows

Jon Warren, Shinzo Watanabe|ArXiv.org|Jul 21, 2003

Stochastic processes and financial applications3 references3 citations

TL;DR

This paper investigates the spectral properties of noise generated by Harris flows—stochastic flows on the real line governed by a correlation function $ b(x) $. Using chaos expansion and Gaussian analysis, it identifies the structure of non-classical noise components beyond finite-order Wiener-Itô chaos, establishing a precise H"older-type regularity estimate for the accessible set of the noise, with the key result showing the Hausdorff dimension of the accessible set is almost surely $ (1-eta)/(2-eta) $, where $ \beta $ is related to the smoothness of $ b(x) $. The analysis relies on time-changed Bessel processes and resolvent estimates for hitting times of zero.

ABSTRACT

We study the noise, in the sense of Tsirelson, generated by Harris flows. A criterion is given for the noise to be non-white, and in this case we study the associated spectral sets.

Motivation & Objective

• To understand the structure of non-classical noise arising from non-strong solutions of stochastic differential equations (SDEs) associated with Harris flows.
• To characterize the spectral properties of the noise generated by Harris flows beyond finite-order Wiener-It\'o chaos expansions.
• To determine the Hausdorff dimension of the accessible set of the noise, which captures the intrinsic irregularity of the flow's noise structure.
• To establish regularity estimates for functionals of the flow in terms of time intervals, using scaling and hitting time analysis of time-changed diffusion processes.

Proposed method

• The noise generated by a Harris flow is analyzed through its associated Gaussian system $ W(t,x) $, constructed as the $ L^2 $-limit of increments of the flow.
• The flow's correlation function $ b(x) $ determines the quadratic variation $ \langle M(x), M(y) \rangle_t = \int_0^t b(X_{0,s}(x) - X_{0,s}(y)) ds $, linking the flow to a continuous martingale structure.
• A time-changed Bessel process $ \widehat{\xi}^+ $ is used to model the evolution of particle distances, with the time change driven by a local time density $ a(\xi) \sim \xi^{\alpha/(1-\alpha)} \wedge 1 $.
• The first hitting time $ \widehat{\sigma}_0 $ of $ \widehat{\xi}^+ $ to zero is analyzed via resolvent densities and scaling properties of Brownian motion.
• Estimates on the probability $ \widehat{P}_\mu(\widehat{\sigma}_0 \leq \epsilon) $ are derived using scaling and bounds on the local time density, leading to $ O(\epsilon^{1/(2-\alpha)}) $ decay.
• The key regularity estimate is obtained by integrating over initial positions and applying a change of measure, leading to the final dimension estimate for the accessible set.

Experimental results

Research questions

• RQ1 What is the structure of the noise generated by a Harris flow beyond finite-order Wiener-It\'o chaos terms?
• RQ2 How does the smoothness of the correlation function $ b(x) $ affect the spectral and geometric properties of the noise?
• RQ3 What is the Hausdorff dimension of the accessible set of the noise generated by a Harris flow?
• RQ4 How does the hitting time distribution of a time-changed Bessel process relate to the regularity of functionals of the flow?
• RQ5 Can the noise generated by a Harris flow be decomposed into classical and non-classical components, and what is the nature of the non-classical part?

Key findings

• The accessible set $ S^{acc}_X $ of the noise generated by a Harris flow has Hausdorff dimension $ (1 - \beta)/(2 - \beta) $ almost surely, where $ \beta $ is related to the H"older regularity of the correlation function $ b(x) $.
• The probability that the accessible set intersects a small interval $[t, t+\epsilon]$ decays as $ O(\epsilon^{1/(2-\alpha)}) $, uniformly in $ t \in [0,1] $, where $ \alpha $ governs the local behavior of $ b(x) $.
• The non-classical component of the noise is characterized by the failure of the Wiener-It\'o chaos expansion to capture all components, with the remainder lying in a singular subspace of $ L^2 $.
• The analysis shows that the noise generated by a Harris flow contains a non-trivial non-classical part, which cannot be represented by a Gaussian white noise or a Poisson noise.
• The spectral structure of the noise is determined by the scaling behavior of the hitting time $ \widehat{\sigma}_0 $ of a time-changed Bessel process, with the decay rate $ \epsilon^{1/(2-\alpha)} $ being optimal.
• The result implies that the noise is not linearizable, and its spectral content lies beyond the classical Wiener-It\'o chaos decomposition, confirming the existence of genuinely non-classical noise in this class of SDEs.