Wave equation with a coloured stable noise

Abstract

Abstract. We define a random measure generated by a real anisotropic harmonizable fractional stable field Z H with stability parameter α ∈ (1, 2) and Hurst index H ∈ (1/2, 1) and prove that the measure is σ-additive in probability. An integral with respect to this measure is constructed, which enables us to consider a wave equation in R 3 with a random source generated by Z H . We show that the solution to this equation, given by Kirchhoff's formula, has a modification, which is Hölder continuous of any order up to (3H − 1) ∧ 1. In the case where H ∈ (2/3, 1), we show further that the modification is absolutely continuous.