Two dimensional delta Bose gas in a weighted space
Sudheesh Surendranath, Li-Cheng Tsai
TL;DR
This work extends the semigroup construction for the two-dimensional delta-Bose gas from the standard $L^{2}$ setting to a weighted $L^{2}$ space $L^{2}_{a}$ that permits exponentially growing functions. The authors develop weighted-norm bounds for the components of the delta-Bose semigroup and its mollified version, and prove strong convergence of the mollified semigroup $\mathcal{Q}^{\varepsilon}(t)$ to the limiting semigroup $\mathcal{Q}(t)$ on $L^{2}_{a}$, with uniform-in-time bounds $\|\mathcal{Q}(t)\|_{2,a\to 2,a} \le c e^{ct}$ and $\|\mathcal{Q}^{\varepsilon}(t)\|_{2,a\to 2,a} \le c e^{ct}$. The analysis hinges on a two-step reduction from weighted to unweighted bounds and a key, nontrivial bound on a Jop-type operator to control the diagrammatic expansion of the semigroup in the weighted setting. This extension enables handling a broader class of initial data and is relevant for studying the Stochastic Heat Flow and its moments, with connections to Gaussian multiplicative chaos and SHF approximations in higher moments.
Abstract
We extend the construction of the semigroup of the two-dimensional delta-Bose gas in Gu, Quastel, and Tsai (2021) (based on Rajeev (1999) and Dimock and Rajeev (2004)) to a weighted $L^2$ space that allows exponentially growing functions. We further show that the semigroup of the mollified delta-Bose gas converges strongly to that of the delta-Bose gas.