Stochastic heat equations driven by space-time $G$-white noise under sublinear expectation

In this paper, we study the stochastic heat equation driven by a multiplicative space-time $G$-white noise within the framework of sublinear expectations. The existence and uniqueness of the mild solution are proved. By generalizing the stochastic Fubini theorem under sublinear expectations, we demonstrate that the mild solution also qualifies as a weak solution. Additionally, we derive moment estimates for the solutions.

💡 Research Summary

The paper develops a comprehensive theory for stochastic heat equations driven by a multiplicative space‑time G‑white noise within the framework of sublinear expectations (also known as G‑expectation). Classical stochastic partial differential equations (SPDEs) are usually formulated under a fixed probability measure, assuming that the driving noise has a known Gaussian distribution. However, many real‑world systems exhibit uncertainty not only in the realization of the noise but also in its underlying probability law. To capture this higher‑order uncertainty, the authors work under a sublinear expectation, which can be represented as the supremum over a family of possibly mutually singular probability measures.

The paper begins by recalling the basic notions of sublinear expectations, including monotonicity, constant preservation, sub‑additivity, positive homogeneity, and regularity. It then introduces the G‑normal distribution, G‑Brownian motion, and, most importantly, spatial and space‑time G‑white noises. Unlike classical white noise, a space‑time G‑white noise is defined via a family of random variables indexed by time‑space rectangles, satisfying a set of consistency, additivity, and independence properties that reflect the sublinear nature of the underlying expectation.

A major technical contribution is the extension of stochastic integration with respect to space‑time G‑white noise. The authors construct the canonical process on a suitable function space, define the filtration, and develop the L^p_G spaces of random fields. They prove that stochastic integrals of deterministic kernels (in particular, the heat kernel) belong to these spaces and satisfy appropriate moment bounds. Crucially, they establish a stochastic Fubini theorem under sublinear expectations, allowing the interchange of time‑space integration order—a tool that is indispensable for relating mild and weak formulations of SPDEs.

The central object of study is the nonlinear stochastic heat equation on a bounded interval