On structure preservation for fully discrete finite difference schemes of stochastic heat equation with L\'evy space-time white noise

TL;DR

This paper develops fully discrete finite difference schemes for the stochastic heat equation with Lévy noise, preserving key solution structures and establishing convergence with near-optimal orders.

Contribution

It introduces schemes that preserve intrinsic solution structures and provides convergence analysis with near-half order in space and quarter order in time.

Findings

Preserves weak intermittency of moments.

Maintains regularity of cdlg paths in negative fractional Sobolev spaces.

Achieves mean-square convergence with orders close to 1/2 in space and 1/4 in time.

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 L\'evy 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 c\`adl\`ag 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.