$G$-Gaussian random fields and stochastic quantization under nonlinear expectation

TL;DR

This paper extends stochastic quantization to the sublinear expectation framework using $G$-Brownian motions, constructing a robust version of the Gaussian free field via SPDEs.

Contribution

It introduces a novel approach for constructing random fields under nonlinear expectations using $G$-stochastic calculus and semigroup methods.

Findings

Derived the unique mild solution to the $G$-Langevin dynamics.

Established the equilibrium distribution as a sublinear expectation analog of the Gaussian free field.

Extended stochastic quantization to the nonlinear expectation setting.

Abstract

We investigate the application of Parisi-Wu stochastic quantization to the construction of random fields within the sublinear expectation framework. Using the semigroup approach and the infinite dimensional $G$-Ornstein Uhlenbeck process, we derive the unique mild solution to the robust Langevin dynamics of bosonic free field -- a parabolic linear stochastic partial differential equation (SPDE) driven by cylindrical $G$-Brownian motions. Mimicking the linear expectation case, we show the equilibrium distribution of the mild solution is the sublinear expectation analog of the massive Gaussian free field.