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

The Kardar-Parisi-Zhang equation (KPZ equation) is solved via Cole-Hopf transfor-mation h = log u, where u is the solution of the multiplicative stochastic heat equa-tion(SHE). In [CD20, CSZ20, G20], they consider the solution of two dimensional KPZ equation via the solution ue of SHE with the flat initial condition and with noise which is mollified in space on scale in epsilon and its strength is weakened as beta e = betaI/ 271-- loge, and they prove that when beta is an element of (0, 1), beta epsilon 1 (log ue - E[log ue]) converges in distribution as a random field to a solution of Edwards-Wilkinson equation. In this paper, we consider a stochastic heat equation ue with a general initial condition u0 and its transformation F(ue) for F in a class of functions F, which contains F(x) = x' (0 < p <= 1) and F(x) = log x. Then, we prove that beta epsilon 1 (F(ue(t, center dot)) - E[F(ue(t,center dot))]) converges in distribution as a random field to a centered Gaussian field jointly in finitely many F is an element of F, t, and u0. In particular, we show the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depend on u0. Our main tools are Ito's formula, the martingale central limit theorem, and the homogenization argument as in [CNN22]. To this end, we also prove a local limit theorem for the partition function of intermediate disorder 2d directed polymers.