[Paper Review] Uniform convergence on subspaces in von Neumann's ergodic theorem with continuous time

A. G. Kachurovskiĭ, I. V. Podvigin+1 more|arXiv (Cornell University)|Feb 27, 2023

Stochastic processes and financial applicationsEconomics, Econometrics and Finance3 citations

TL;DR

This paper establishes spectral criteria for power uniform convergence in von Neumann's ergodic theorem with continuous time, focusing on convergence on subspaces with their own norms. It characterizes all possible convergence exponents α ∈ [0,2], provides complete descriptions of subspaces exhibiting such convergence, and refines prior estimates on convergence rates for (semi)flows, showing uniform convergence on the full Hilbert space occurs only in trivial cases with spectral gaps.

ABSTRACT

Power-law uniform (in the operator norm) convergence on vector subspaces with their own norms in von Neumann's ergodic theorem with continuous time is considered. All possible exponents of the considered power-law convergence are found; for each of these exponents, spectral criteria for such convergence are given and a complete description of all such subspaces is obtained. Uniform convergence over the entire space takes place only in trivial cases, which explains the interest in the uniform convergence just on subspaces. In addition, along the way, the old convergence rate estimates in the von Neumann ergodic theorem for (semi)flows are generalized and refined.

Motivation & Objective

• To characterize all possible exponents α ∈ [0,2] for which power uniform convergence occurs in von Neumann's ergodic theorem with continuous time.
• To derive spectral criteria for power uniform convergence on subspaces with their own norms.
• To provide a complete description of all vector subspaces (with their own norms) exhibiting power uniform convergence with any exponent α ∈ [0,2].
• To refine and generalize prior estimates on convergence rates in the von Neumann ergodic theorem for (semi)flows.
• To explain why uniform convergence on the entire Hilbert space is rare, occurring only in trivial cases with spectral gaps.

Proposed method

• The analysis uses the spectral measure σ_f associated with a vector f ∈ H and the Fejér kernel F_τ(x) = (sin(τx/2)/(τx/2))² to express the L²-norm of the ergodic average P_{t,s}f − f*.
• A key technical tool is integration by parts, inspired by Gapsky's method, to derive sharp L²-estimates for the convergence rate.
• The paper establishes a spectral criterion for power uniform convergence on one-dimensional subspaces by analyzing the decay rate of the spectral measure σ_f−f* near zero.
• For multidimensional subspaces, the method extends the one-dimensional criteria by considering the joint behavior of spectral measures across basis vectors.
• The convergence rate is analyzed via the integral ∫ℝ F_τ(x) dσ_f−f*(x), and the decay of this integral as τ = t−s → ∞ determines the exponent α.
• The results are extended to Banach subspaces continuously and densely embedded in H, with operator norm estimates of order O(t^{−l}) for some l > 0.

Experimental results

Research questions

• RQ1 What are all possible exponents α ∈ [0,2] for which power uniform convergence occurs in von Neumann's ergodic theorem with continuous time on subspaces with their own norms?
• RQ2 What spectral conditions on the spectral measure σ_f must be satisfied for a given α ∈ [0,2] to ensure power uniform convergence on a one-dimensional subspace?
• RQ3 How can one completely characterize all finite-dimensional subspaces (with their own norms) that exhibit power uniform convergence with a given exponent α ∈ [0,2]?
• RQ4 Why is uniform convergence on the entire Hilbert space H only possible in trivial cases, and what spectral condition (spectral gap) characterizes such cases?
• RQ5 How do the new estimates for convergence rates refine and improve upon earlier results in the literature for (semi)flows?

Key findings

• All possible exponents of power uniform convergence are precisely the values α ∈ [0,2], with α = 2 being the maximal achievable rate.
• For each α ∈ [0,2], a spectral criterion is established: power uniform convergence holds on a one-dimensional subspace if and only if the spectral measure σ_f satisfies ∫_{|x|≤ε} dσ_f(x) = O(ε^{2−α}) as ε → 0.
• The maximal convergence rate α = 2 is achieved if and only if the spectral measure σ_f satisfies ∫_{|x|≤ε} dσ_f(x) = O(ε^0) = O(1) as ε → 0, i.e., the measure has a bounded density near zero.
• A complete description of all finite-dimensional subspaces with power uniform convergence is obtained: such a subspace exists with exponent α if and only if the spectral measures of its basis vectors satisfy the corresponding decay condition.
• The paper provides a sharp constant in the rate estimate: the constant in the O(t^{−α}) bound is optimal, improving upon earlier estimates in the literature.
• Uniform convergence on the entire Hilbert space H holds only in the case of a spectral gap at zero, which corresponds to periodic trajectories in the underlying dynamical system, a rare and trivial case.