Almost sure CLT for hyperbolic Anderson model with Lévy colored noise

In this note, we prove the Almost Sure Central Limit Theorem (ASCLT) for the spatial integral of the solution of the hyperbolic Anderson model driven by the Lévy colored noise introduced in Balan (2015). For this, we use the central limit theorem for the normalized spatial integral, and an estimate for the Malliavin derivative of the solution, both derived in the recent preprint Balan and Stephenson (2026). We assume that the spatial correlation kernel of the noise is either integrable, or it is given by the Riesz kernel.

💡 Research Summary

The paper establishes an Almost Sure Central Limit Theorem (ASCLT) for the spatial integral of the solution to the one‑dimensional hyperbolic Anderson model driven by Lévy colored noise. The model is given by
∂²_t u(t,x)=∂²_x u(t,x)+u(t,x)·Ẋ(t,x), t>0, x∈ℝ,
with initial conditions u(0,x)=1 and ∂t u(0,x)=0. The driving noise Ẋ is the Lévy colored noise introduced by Balan (2015), defined through a Poisson random measure N on ℝ₊×ℝ×ℝ₀ with intensity dt dx ν(dz). The spatial correlation kernel κ of the noise is assumed either integrable (κ∈L¹(ℝ)) or a Riesz kernel R{1,α/2} for some α∈(0,1).

The authors rely on two key results proved in a recent preprint (Balan & Stephenson, 2026): (i) the variance of the spatial average F_R(t)=∫_{-R}^{R}