Spectrums and Uniform Mean Ergodicity of Weighted Composition Operators on Fock Spaces

For holomorphic pairs of symbols $$(u, \psi )$$ ( u , ψ ) , we study various structures of the weighted composition operator $$ W_{(u,\psi )} f= u \cdot f(\psi )$$ W ( u , ψ ) f = u · f ( ψ ) defined on the Fock spaces $$\mathcal {F}_p$$ F p . We have identified operators $$W_{(u,\psi )}$$ W ( u , ψ ) that have power-bounded and uniformly mean ergodic properties on the spaces. These properties are described in terms of easy to apply conditions relying on the values |u(0)| and $$|u(\frac{b}{1-a})|$$ | u ( b 1 - a ) | , where a and b are coefficients from linear expansion of the symbol $$\psi $$ ψ . The spectrum of the operators is also determined and applied further to prove results about uniform mean ergodicity.

Dynamics of Weighted Composition Operators on Spaces of Entire Functions of Exponential and Infraexponential Type

Given an affine symbol $$\varphi $$ φ and a multiplier w, we focus on the weighted composition operator $$C_{w, \varphi }$$ C w , φ acting on the spaces Exp and $$Exp^0$$ E x p 0 of entire functions of exponential and of infraexponential type, respectively. We characterize the continuity of the operator and, for w the product of a polynomial by an exponential function, we completely characterize power boundedness and (uniform) mean ergodicity. In the case of multiples of composition operators, we also obtain the spectrum and characterize hypercyclicity.

Dynamic characterization of a stochastic SIR infectious disease model with dual perturbation

Environmental perturbations are unavoidable in the propagation of infectious diseases. In this paper, we introduce the stochasticity into the susceptible–infected–recovered (SIR) model via the parameter perturbation method. The stochastic disturbances associated with the disease transmission coefficient and the mortality rate are presented with two perturbations: Gaussian white noise and Lévy jumps, respectively. This idea provides an overview of disease dynamics under different random perturbation scenarios. By using new techniques and methods, we study certain interesting asymptotic properties of our perturbed model, namely: persistence in the mean, ergodicity and extinction of the disease. For illustrative purposes, numerical examples are presented for checking the theoretical study.

THE CESÀRO OPERATOR IN THE FRÉCHET SPACES ℓp+ AND Lp−

The classical spaces ℓp+, 1 ≤ p < ∞, and Lp−, 1<p ≤ ∞, are countably normed, reflexive Fréchet spaces in which the Cesàro operator C acts continuously. A detailed investigation is made of various operator theoretic properties of C (e.g., spectrum, point spectrum, mean ergodicity) as well as certain aspects concerning the dynamics of C (e.g., hypercyclic, supercyclic, chaos). This complements the results of [3, 4], where C was studied in the spaces ℂℕ, Lploc(ℝ+) for 1 < p < ∞ and C(ℝ+), which belong to a very different collection of Fréchet spaces, called quojections; these are automatically Banach spaces whenever they admit a continuous norm.

Toward a practical approach for ergodicity analysis

It is of importance to perform hydrological forecast using a finite hydrological time series. Most time series analysis approaches presume a data series to be ergodic without justifying this assumption. This paper presents a practical approach to analyze the mean ergodic property of hydrological processes by means of autocorrelation function evaluation and Augmented Dickey Fuller test, a radial basis function neural network, and the definition of mean ergodicity. The mean ergodicity of precipitation processes at the Lanzhou Rain Gauge Station in the Yellow River basin, the Ankang Rain Gauge Station in Han River, both in China, and at Newberry, MI, USA are analyzed using the proposed approach. The results indicate that the precipitations of March, July, and August in Lanzhou, and of May, June, and August in Ankang have mean ergodicity, whereas, the precipitation of any other calendar month in these two rain gauge stations do not have mean ergodicity. The precipitation of February, May, July, and December in Newberry show ergodic property, although the precipitation of each month shows a clear increasing or decreasing trend.