Approximation of the L\'evy-driven stochastic heat equation on the sphere
Abstract
The stochastic heat equation on the sphere driven by additive L\'evy random field is approximated by a spectral method in space and forward and backward Euler-Maruyama schemes in time, in analogy to the Wiener case. New regularity results are proven for the stochastic heat equation. The spectral approximation is based on a truncation of the series expansion with respect to the spherical harmonic functions. To do so, we restrict to square-integrable random field and optimal strong convergence rates for a given regularity of the initial condition and two different settings of regularity for the driving noise are derived for the Euler-Maruyama methods. Besides strong convergence, convergence of the expectation and second moment is shown. Weak rates for the spectral approximation are discussed. Numerical simulations confirm the theoretical results.
Keywords
Cite
@article{arxiv.2507.05005,
title = {Approximation of the L\'evy-driven stochastic heat equation on the sphere},
author = {Annika Lang and Andrea Papini and Verena Schwarz},
journal= {arXiv preprint arXiv:2507.05005},
year = {2025}
}