G-Gaussian processes under sublinear expectations and $ q $-Brownian motion in quantum mechanics

Shige Peng

doi:10.3934/naco.2022034

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2023, Volume 13, Issue 3&4: 583-603. Doi: 10.3934/naco.2022034

This issue Previous Article A level-set approach to the control of state-constrained McKean-Vlasov equations: application to renewable energy storage and portfolio selection Next Article Approximation of

$ N $

-player stochastic games with singular controls by mean field games

G-Gaussian processes under sublinear expectations and $ q $-Brownian motion in quantum mechanics

Shige Peng

School of Mathematics, Shandong University, 250100 China

Received: February 2022

Revised: October 2022

Early access: November 2022

Published: September 2023

This paper is partially supported by [NSFC1114005132101]

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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].

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Mathematics Subject Classification: Primary: 60A, 60H10, 60H05, 60H30, 60H05, 60H30, 60E05.

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