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

Xiaojun Ji, Shige Peng

TL;DR

The paper addresses stochastic heat equations under probability model uncertainty by formulating them in a sublinear $G$-expectation framework with space-time $G$-white noise. It develops stochastic integration and a stochastic Fubini theorem under sublinear expectations, and proves existence and uniqueness of mild solutions for nonlinear equations, with mild solutions also qualifying as weak solutions and accompanied by moment estimates. The linear case yields explicit mild/weak solutions via the heat kernel, while the nonlinear case is handled by Picard iteration within $\mathbf{S}_G^2$ spaces, establishing regularity and interdependence between formulations. These results provide a rigorous foundation for SPDEs under distributional ambiguity, with applications to systems subject to probability uncertainty in physics and finance.

Abstract

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.

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

TL;DR

The paper addresses stochastic heat equations under probability model uncertainty by formulating them in a sublinear -expectation framework with space-time -white noise. It develops stochastic integration and a stochastic Fubini theorem under sublinear expectations, and proves existence and uniqueness of mild solutions for nonlinear equations, with mild solutions also qualifying as weak solutions and accompanied by moment estimates. The linear case yields explicit mild/weak solutions via the heat kernel, while the nonlinear case is handled by Picard iteration within spaces, establishing regularity and interdependence between formulations. These results provide a rigorous foundation for SPDEs under distributional ambiguity, with applications to systems subject to probability uncertainty in physics and finance.

Abstract

In this paper, we study the stochastic heat equation driven by a multiplicative space-time -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.
Paper Structure (10 sections, 15 theorems, 135 equations)

This paper contains 10 sections, 15 theorems, 135 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Basic notions of sublinear expectations
  4. $G$-white noises under sublinear expectations
  5. Stochastic integrals with respect to space-time $G$-white noises
  6. Stochastic Fubini Theorem
  7. Stochastic heat equations driven by space-time $G$-white noises
  8. Linear stochastic heat equations
  9. Nonlinear stochastic heat equations
  10. Applications to problems with probability uncertainty

Key Result

Theorem 2.11

(See DHPHPJP) There exists a weakly compact set $\mathcal{P}$ of probability measures on $(\Omega,\mathcal{B}(\Omega))$ such that where $\mathcal{B}(\Omega)$ is the $\sigma$-algebra generated by all open sets. $\mathcal{P}$ is called the family of probability measures that represents $\hat{\mathbb{E}}$.

Theorems & Definitions (35)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 25 more