Vol. 2, No. 1, 2021

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Moments of the 2D SHE at criticality

Yu Gu, Jeremy Quastel and Li-Cheng Tsai

Vol. 2 (2021), No. 1, 179–219

We study the stochastic heat equation in two spatial dimensions with a multiplicative white noise, as the limit of the equation driven by a noise that is mollified in space and white in time. As the mollification radius 𝜀 0, we tune the coupling constant near the critical point, and show that the single time correlation functions converge to a limit written in terms of an explicit nontrivial semigroup. Our approach consists of two steps. First we show the convergence of the resolvent of the (tuned) two-dimensional delta Bose gas, by adapting the framework of Dimock and Rajeev (J. Phys. A 37:39 (2004), 9157–9173) to our setup of spatial mollification. Then we match this to the Laplace transform of our semigroup.

stochastic heat equation, delta Bose gas, two-dimensional, critical
Mathematical Subject Classification
Primary: 60H15
Secondary: 35R60, 82D60, 46N30
Received: 5 April 2020
Revised: 22 September 2020
Accepted: 11 October 2020
Published: 16 March 2021
Yu Gu
Department of Mathematics
Carnegie Mellon University
Pittsburgh PA
United States
Jeremy Quastel
Department of Mathematics
University of Toronto
Toronto ON
Li-Cheng Tsai
Department of Mathematics
Rutgers University
New Brunswick
Piscataway NJ
United States