Approximation of the Lévy-driven stochastic heat equation on the sphere

Preprint, 2025

The stochastic heat equation on the sphere driven by additive L´evy random fieldis approximated by a spectral method in space and forward and backward Euler–Maruyamaschemes in time, in analogy to the Wiener case. New regularity results are proven for thestochastic heat equation. The spectral approximation is based on a truncation of the seriesexpansion 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 theinitial condition and two different settings of regularity for the driving noise are derived forthe Euler–Maruyama methods. Besides strong convergence, convergence of the expectationand second moment is shown. Weak rates for the spectral approximation are discussed.Numerical simulations confirm the theoretical results.