January 2024 Delta-Bose gas from the viewpoint of the two-dimensional stochastic heat equation
Yu-Ting Chen
Author Affiliations +
Ann. Probab. 52(1): 127-187 (January 2024). DOI: 10.1214/23-AOP1649

Abstract

In order to give a dual, annealed description of the two-dimensional stochastic heat equation (SHE) from regularizing the noise, we consider the Schrödinger semigroup of the many-body delta-Bose gas from mollifying the delta potentials. The main theorem proves the convergences of the corresponding approximate semigroups when they act on bounded functions. For the proof we introduce a mean-field Poisson system to expand the Feynman–Kac formula of the approximate semigroups. This expansion yields infinite series, showing certain Markovian decompositions of the summands into nonconcurrent, nonconsecutive two-body interactions. Components in these decompositions are then grouped nonlinearly in time to establish the dominated convergence of the infinite series. With regards to the two-dimensional SHE, the main theorem also characterizes the Nth moments for all N3 under any bounded initial condition. A particular example is the constant initial condition under which the solution of the SHE has the interpretation of the partition function of the continuum directed random polymer.

Funding Statement

Support from the Natural Science and Engineering Research Council of Canada is gratefully acknowledged.

Acknowledgments

The author would like to thank an anonymous referee and an anonymous Associate Editor for their constructive comments. Special thanks go to Rongfeng Sun for suggesting that the present model be considered.

Citation

Download Citation

Yu-Ting Chen. "Delta-Bose gas from the viewpoint of the two-dimensional stochastic heat equation." Ann. Probab. 52 (1) 127 - 187, January 2024. https://doi.org/10.1214/23-AOP1649

Information

Received: 1 February 2022; Revised: 1 July 2023; Published: January 2024
First available in Project Euclid: 29 January 2024

Digital Object Identifier: 10.1214/23-AOP1649

Subjects:
Primary: 60J55 , 60J65
Secondary: 60H30

Keywords: additive functional , Delta-Bose gas , diagrammatic expansion , Directed polymer , Feynman–Kac formula , Kardar–Parisi–Zhang equation , Planar Brownian motion , Stochastic heat equation

Rights: Copyright © 2024 Institute of Mathematical Statistics

Vol.52 • No. 1 • January 2024
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