Fluctuations of two-dimensional stochastic heat equation and KPZ equation in subcritical regime for general initial conditions

Abstract

The solution of Kardar-Parisi-Zhang equation (KPZ equation) is solved formally via Cole-Hopf transformation $h=\log u$, where $u$ is the solution of multiplicative stochastic heat equation(SHE). In earlier works by Chatterjee and Dunlap, Caravenna, Sun, and Zygouras, and Gu, they consider the solution of two dimensional KPZ equation via the solution $u_\varepsilon$ of SHE with flat initial condition and with noise which is mollified in space on scale in $\varepsilon$ and its strength is weakened as $\beta_\varepsilon=\hat{\beta} \sqrt{\frac{2\pi \varepsilon}{-\log \varepsilon}}$, and they prove that when $\hat{\beta}\in (0,1)$, $\frac{1}{\beta_\varepsilon}(\log u_\varepsilon-\mathbb{E}[\log u_\varepsilon])$ converges in distribution to a solution of Edward-Wilkinson model as a random field. In this paper, we consider a stochastic heat equation $u_\varepsilon$ with general initial condition $u_0$ and its transformation $F(u_\varepsilon)$ for $F$ in a class of functions $\mathfrak{F}$, which contains $F(x)=x^p$ ($0<p\leq 1$) and $F(x)=\log x$. Then, we prove that $\frac{1}{\beta_\varepsilon}(F(u_\varepsilon(t,x))-\mathbb{E}[F(u_\varepsilon(t,x))])$ converges in distribution to Gaussian random variables jointly in finitely many $F\in \mathfrak{F}$, $t$, and $u_0$. In particular, we obtain the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depends on $u_0$. Our main tools are It\^o's formula, the martingale central limit theorem, and the homogenization argument as in the works by Cosco and the authors. To this end, we also prove the local limit theorem for the partition function of intermediate $2d$-directed polymers