Central limit theorem for the focusing Φ⁴-measure in the infinite volume limit

2024/11/12 by Kihoon Seong, Philippe Sosoe, Seong, Kihoon +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Navier-Stokes equation solutions #Probability (math.PR)

paper · pdf · doi:10.48550/arxiv.2411.07840

openalex publication_date 2024/11/12 · openalex created_date 2024/11/15 · openalex updated_date 2026/07/29

Abstract

We study the fluctuations of the focusing Φ⁴-measure on the one-dimensional torus in the infinite volume limit. This measure is an invariant Gibbs measure for the nonlinear Schrödinger equation. It had previously been shown by B. Rider that the measure is strongly concentrated around a family of minimizers of the Hamiltonian associated with the measure. These exhibit increasingly sharp spatial concentration, resulting in a trivial limit to first order. We study the fluctuations around this soliton manifold. We show that the scaled field under the Gibbs measure converges to white noise in the limit, identifying the next order fluctuations predicted by B. Rider.

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