2025/04/08 by Francesco Caravenna, Caravenna, Francesco, Rongfeng Sun +3 · 4 citations
Mathematics · #60H15 #82D60 #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Physics (math-ph) #Primary: 82B44 #Probability (math.PR) #Secondary: 35R60 #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2504.06128
openalex publication_date 2025/04/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The Critical 2D Stochastic Heat Flow (SHF) provides a natural candidate solution to the ill-posed 2D Stochastic Heat Equation with multiplicative space-time white noise. In this paper, we initiate the investigation of the spatial properties of the SHF. We prove that, as a random measure on ℝ2, it is a.s. singular w.r.t. the Lebesgue measure. This is obtained by probing a "quasi-critical" regime and showing the asymptotic log-normality of the mass assigned to vanishing balls, as the disorder strength is sent to zero at a suitable rate, accompanied by similar results for critical 2D directed polymers. We also describe the regularity of the SHF, showing that it is a.s. Hölder C-ε for any ε>0, implying the absence of atoms, and we establish local convergence to zero in the long time limit.