2025/04/30 by Yu-Ting Chen, Chen, Yu-Ting
Mathematics · #FOS: Mathematics #Nonlinear Partial Differential Equations #Probability (math.PR) #Spectral Theory in Mathematical Physics #Stochastic processes and statistical mechanics
paper · pdf · doi:10.48550/arxiv.2504.21796
openalex publication_date 2025/04/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
By taking the viewpoint of Brownian additive functionals, we extend an existing approximation theorem of the two-dimensional Laplacian singularly perturbed at the origin. The approximate operators are defined by adding a rescaled function to the two-dimensional Laplacian such that the support of the perturbation thus produced converges to a point in the limit of the rescaling, and suitable renormalizations of the coupling constants are also imposed. The first main theorem assumes the zero total mass of the function for rescaling. It proves the limits of the approximate resolvents up to an extended criticality defined by two levels of phase transitions. The second main theorem handles the case of positive total mass of the function for rescaling and proves an asymptotic expansion of a multivariate density induced by the approximate resolvent. Moreover, this expansion is valid beyond criticality.