G-GAUSSIAN PROCESSES UNDER SUBLINEAR EXPECTATIONS AND q-BROWNIAN MOTION IN QUANTUM MECHANICS

An important observation of this paper is that a non-trivial G-Brownian motion is not a Gaussian process, e.g., finite dimensional distributions of G-Brownian motion is not G-normal, or G-Gaussian. We then have to start from the very beginning, to establish the foundation of G-Gaussian processes which is more suitable for space-parameter systems. It is known that the notion of classical Brownian motion is not suitable to model the random propagation of a quantum particle. In this paper we have rigorously defined a new stochastic process called q-Brownian who's propagator exactly coincides with the one proposed by Feynman based on the solution of Schrodinger equation. The notion of expectation plays a fundamental base of the above results. This paper was originally published on [38].