analysis of partial differential equations

The probabilistic convergence problem of density functions related to

arXiv:2503.13561

summary

The paper proves that density functions associated with the operator ∂ₓ³+∂ₓ⁻¹ on the real line converge probabilistically, using full randomization, high‑low frequency decomposition, and Schatten‑class (𝔖²) techniques.

Abstract

In this article, by using full randomization introduced by Hadama and Yamamoto (Probabilistic Strichartz estimates in Schatten classes and their applications to Hartree equation, J. Math. Phys. 67(2026), 35pp) and high-low frequency technique as well as the property of , we establish the probabilistic convergence of the density function related to on , which extends the Theorem 1.3 of Yan et al. (Convergence problem of Ostrovsky equation with rough data and random data, Indiana Univ. Math. J. 71(2022), 1897-1921.).

Topics & keywords

#probabilistic PDE#density function convergence#randomization methods#high‑low frequency technique#Schatten class#Ostrovsky equation∂ₓ³+∂ₓ⁻¹probabilistic Strichartz estimates𝔖²full randomizationhigh‑low frequency decompositiondensity function