Low temperature properties in the stochastic quantization of Brownian oscillator

The microscopic model of Ullersma (1966) for a harmonic oscillator in contact with a thermal reservoir is quantised in the framework of Nelson's stochastic mechanics. Eliminating the degrees of freedom of the thermal reservoir the stochastic process of the quantum Brownian oscillator is obtained as the sum of the thermal contribution and the quantum (zero-temperature) contribution. The properties of the quantum fluctuations are studied in detail and the equation of motion is derived, obtaining the first example of a non-Markovian Nelson process