On structure preservation for fully discrete finite difference schemes of stochastic heat equation with Lévy space-time white noise

TL;DR

The paper tackles structure preservation in numerical schemes for the stochastic heat equation driven by Lévy space-time white noise. It introduces a fully discrete finite difference scheme (spatial discretization plus a $\theta$-time scheme) and analyzes its ability to retain the exact solution's weak intermittency and càdlàg path properties through discrete Green-function estimates. A key contribution is establishing mean-square convergence of order nearly $\tfrac{1}{2}$ in space and $\tfrac{1}{4}$ in time, with additional results for infinite-variance noise via truncation. The work further shows tightness and weak relative compactness in appropriate function spaces, clarifying the numerical approximation's reliability for non-Gaussian SPDEs with jumps and providing pathways to almost sure convergence under noise truncation. These results have practical implications for robust simulations of stochastic systems with jumps, including financial, physical, and biological models.

Abstract

This paper investigates the structure preservation and convergence analysis of a class of fully discrete finite difference schemes for the stochastic heat equation driven by Lévy space-time white noise. The novelty lies in the simultaneous preservation of intrinsic structures for the exact solution, in particular the weak intermittency of moments and the regularity of càdlàg path in negative fractional Sobolev spaces. The key in the proof is the detailed analysis of technical estimates for discrete Green functions of the numerical solution. This analysis is also crucial in establishing the mean-square convergence of the schemes with orders of almost $\frac12$ in space and almost $\frac14$ in time.

Theorems & Definitions (18)

• Proposition 2.1
• Proposition 2.2
• proof
• Proposition 2.3
• Lemma 3.1
• proof
• Lemma 3.2: Inverse Grönwall inequality
• proof
• Theorem 3.3
• proof