The fractional oscillator process with two indices

• Published in: Journal of physics. A, Mathematical and theoretical Vol. 42; no. 6; pp. 065208 - 065208 (34)
• Main Authors: Lim, S C, Teo, L P
• Format: Journal Article
• Language: English
• Published: Bristol IOP Publishing 13.02.2009; IOP
• Subjects: Exact sciences and technology; Physics; Fluctuations; Quantization; Stochastic method; Oscillators; Covariance; Fractal dimension; Differential equations; Zeta function; Diffusion process; Diffusion; Klein Gordon field; Regularization; Stochastic equation
• ISSN: 1751-8121; 1751-8113; 1751-8121
• DOI: 10.1088/1751-8113/42/6/065208

We introduce a new fractional oscillator process which can be obtained as a 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. By considering the fractional oscillator process as the velocity of a diffusion process, we derive the corresponding diffusion constant, fluctuation-dissipation relation and mean-square displacement. The fractional oscillator process can also be regarded as a 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.