  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.
