Fractional oscillator process with two indices

TL;DR

This paper introduces a novel fractional oscillator process characterized by two fractional orders, analyzing its properties, physical interpretation as a fractional field, and calculating associated Casimir energy using advanced mathematical techniques.

Contribution

It presents a new fractional oscillator process with two fractional indices, exploring its properties and physical implications, including a novel approach to Casimir energy calculation.

Findings

Process exhibits fractal dimension and short-range dependence.

Can be modeled as a fractional Euclidean Klein-Gordon field.

Casimir energy computed via zeta function regularization.

Abstract

We introduce a new fractional oscillator process which can be obtained as solution of a stochastic differential equation with two fractional orders. Basic properties such as fractal dimension and short range dependence of the process are studied by considering the asymptotic properties of its covariance function. The fluctuation--dissipation relation of the process is investigated. The fractional oscillator process can be regarded as one-dimensional fractional Euclidean Klein-Gordon field, which can be obtained by applying the Parisi-Wu stochastic quantization method to a nonlocal Euclidean action. The Casimir energy associated with the fractional field at positive temperature is calculated by using the zeta function regularization technique.