G-Gaussian Processes under Sublinear Expectations and q-Brownian Motion in Quantum Mechanics

TL;DR

This paper develops a framework for constructing generalized Gaussian processes and q-Brownian motion within nonlinear and complex-valued expectation spaces, enabling new representations of quantum mechanical equations.

Contribution

It introduces a novel method to build stochastic processes under nonlinear expectations and applies it to quantum mechanics, deriving a new Feynman-Kac formula for Schrödinger equations.

Findings

Constructed a generalized Gaussian process under sublinear expectation.

Developed a q-Brownian motion under complex-valued expectation.

Derived a new Feynman-Kac formula for quantum Schrödinger equations.

Abstract

We provide a general approach to construct a stochastic process with a given consistent family of finite dimensional distributions under a nonlinear expectation space. We use this approach to construct a generalized Gaussian process under a sublinear expectation and a q-Brownian motion. The later one is under a complex-valued linear expectation, with which a new type of Feynman-Kac formula can be derived to represent the solution of a Schr\"odinger equation.