Quantum Gaussian Processes

This paper studies construction of quantum Gaussian processes based on ordinary Gaussian processes through their reproducing kernel Hilbert spaces, and investigate the relationship between the stochastic properties of the quantum Gaussian processes and the base Gaussian processes. In particular, we construct quantum Brownian bridges and quantum Ornstein-Uhlenbeck processes. Non-commutative stochastic calculus for operator-valued processes has been studied by many authors ((1, 7, 8, 9]). In 13], P. A. Meyer, using Brownian motion and Poisson processes, gave the probabilistic explanation of the Fock spaces. In this paper, we will construct quantum Gaussian processes based on ordinary Gaussian processes through their reproducing kernel Hilbert spaces, and investigate the relationship between the stochastic properties of the quantum Gaussian processes and the base Gaussian processes.