[Paper Review] On spectral asymptotics of the tensor product of operators with almost regular marginal asymptotics

Nikita Rastegaev|arXiv (Cornell University)|Mar 31, 2018

advanced mathematical theories14 references3 citations

TL;DR

This paper generalizes spectral asymptotics for tensor products of compact self-adjoint operators with almost regular marginal eigenvalue behavior, showing that under mild conditions the tensor product inherits almost regular asymptotics. The key result establishes precise logarithmic asymptotics for small ball probabilities of Gaussian random fields, extending prior work on Brownian sheets and related processes with fractal measures.

ABSTRACT

Spectral asymptotics of a tensor product of compact operators in Hilbert space with known marginal asymptotics is studied. Methods of A. Karol', A. Nazarov and Ya. Nikitin (Trans. AMS, 2008) are generalized for operators with almost regular marginal asymptotics. In many (but not all) cases it is shown, that tensor product has almost regular asymptotics as well. Obtained results are then applied to the theory of small ball probabilities of Gaussian random fields.

Motivation & Objective

• To extend the spectral asymptotics theory for tensor products of compact operators beyond regular asymptotics to operators with almost regular marginal eigenvalue behavior.
• To determine under which conditions the tensor product of two operators with almost regular eigenvalue asymptotics also exhibits almost regular spectral behavior.
• To apply the derived spectral asymptotics to the problem of small ball probabilities for Gaussian random fields, particularly those with self-similar or fractal measures.
• To generalize prior results on Brownian sheets and Green processes to broader classes of Gaussian fields with periodic fluctuations in their spectral counting functions.

Proposed method

• Generalizes methods from Karol, Nazarov, and Nikitin (2008) to handle operators with eigenvalues asymptotically of the form $\lambda_n \sim \frac{\psi(n) \cdot \mathfrak{s}(\ln n)}{n^p}$, where $\psi$ is slowly varying and $\mathfrak{s}$ is periodic.
• Analyzes the asymptotic behavior of the counting function $\mathcal{N}(t, \mathcal{T} \otimes \widetilde{\mathcal{T}})$ as $t \to 0^+$, using convolutions of almost Mellin-type integrals.
• Classifies the resulting asymptotics into cases based on the relative power parameters $p$ and $\widetilde{p}$, and the convergence of integrals involving the slowly varying functions.
• Uses the equivalence between eigenvalue asymptotics and counting function asymptotics: $\mathcal{N}(t) \sim \frac{\varphi(1/t) \cdot s(\ln(1/t))}{t^{1/p}}$, where $\varphi$ is slowly varying and $s$ is periodic.
• Applies the spectral results to Gaussian fields via the connection between the spectral asymptotics of the covariance operator and the logarithmic small ball probabilities.
• Derives explicit asymptotic forms for $\ln \mathbf{P}(\|Z\|_{\mu} \leq \varepsilon)$ as $\varepsilon \to 0$, involving periodic functions and power-logarithmic corrections.

Experimental results

Research questions

• RQ1 Under what conditions does the tensor product of two compact operators with almost regular eigenvalue asymptotics inherit almost regular spectral behavior?
• RQ2 How do the relative power parameters $p$ and $\widetilde{p}$, and the periodicity of the spectral fluctuations, affect the asymptotic structure of the tensor product's eigenvalue counting function?
• RQ3 What is the precise form of the small ball probability asymptotics for Gaussian random fields whose covariance operators are tensor products of operators with almost regular spectral behavior?
• RQ4 How do incommensurable periods or degenerate periodic components influence the spectral asymptotics of the tensor product?

Key findings

• The tensor product of two operators with almost regular asymptotics $\lambda_n \sim \frac{\psi(n) \cdot \mathfrak{s}(\ln n)}{n^p}$ inherits almost regular spectral asymptotics in cases where $\widetilde{p} > p$ or $\widetilde{p} = p$ with divergent integrals and commensurate periods.
• When the periods of the periodic components are incommensurate, the tensor product's spectral asymptotics become regular (i.e., the periodic component degenerates to a constant).
• In cases where one or both integrals $\int_1^\infty \varphi(\sigma) \frac{d\sigma}{\sigma}$ converge, the resulting asymptotics are more complex and involve logarithmic corrections.
• For the Brownian sheet with generalized Cantor measures, the small ball probability satisfies $\ln \mathbf{P}(\|\mathbb{W}_d\|_{\mu} \leq \varepsilon) \sim -\varepsilon^{-2\log_3 2} \ln^{(d-1)\log_3 6}(1/\varepsilon) \zeta(\ln(1/\varepsilon))$, where $\zeta$ is a $\frac{\ln 3}{2}$-periodic function.
• The asymptotic form of the small ball probability depends on the number $\mathfrak{d}$ of operators with the minimal power exponent $\mathfrak{p}$, and the periodic component of the tensor product is obtained by iteratively applying a convolution formula.