G-Gaussian processes under sublinear expectations and <inline-formula><tex-math id="M1">$ q $</tex-math></inline-formula>-Brownian motion in quantum mechanics

Abstract

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 Schrödinger equation. The notion of expectation plays a fundamental base of the above results. This paper was originally published on [38].