Abstract

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 $𝜀\to 0$, 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.

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