Approximation of the L\'evy-driven stochastic heat equation on the sphere

TL;DR

This paper develops a spectral approximation method for the Le9vy-driven stochastic heat equation on the sphere, establishing regularity results and convergence rates, with numerical simulations validating the theoretical findings.

Contribution

It introduces a spectral approximation combined with Euler-Maruyama schemes for the Le9vy-driven stochastic heat equation on the sphere, providing new regularity and convergence results.

Findings

Spectral approximation converges strongly with optimal rates.

Expectation and second moment convergence are established.

Numerical simulations confirm theoretical convergence rates.

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.