Martingale problem of the two-dimensional stochastic heat equation at criticality

2025/04/30 by Yu-Ting Chen, Chen, Yu-Ting · 6 citations
Economics, Econometrics and Finance · Mathematics · #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Probability (math.PR) #Stochastic processes and financial applications #Stochastic processes and statistical mechanics

paper · pdf · doi:10.48550/arxiv.2504.21791

openalex publication_date 2025/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We study the martingale formulation of the two-dimensional stochastic heat equation (SHE) at criticality. The main theorem proves an exact recursive-type equation that expresses the covariation measures of the SHE in terms of the solutions via an integro-multiplication operator. As an application, the quadratic variations of the martingale parts in the mild form are proven explicitly expressible in the solutions of the SHE and the two-dimensional two-body delta-Bose gas semigroups. The proofs are based on the standard approximations of the two-dimensional SHE at criticality, and now we analyze asymptotic expansions of the covariation measures of the approximate solutions in the limit. Also, new bounds for certain mixed moments of the fourth order of the approximate solutions are among the main tools for a priori estimates.

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