## Abstract

The Kardar-Parisi-Zhang equation (KPZ equation) is solved via Cole-Hopf transformation $h=logu$, where *u* is the solution of the multiplicative stochastic heat equation(SHE). In [CD20, CSZ20, G20], they consider the solution of two dimensional KPZ equation via the solution ${u}_{\mathit{\epsilon}}$ of SHE with the flat initial condition and with noise which is mollified in space on scale in *ε* and its strength is weakened as ${\mathit{\beta}}_{\mathit{\epsilon}}=\stackrel{\u02c6}{\mathit{\beta}}\sqrt{\frac{2\mathrm{\pi}}{-log\mathit{\epsilon}}}$, and they prove that when $\stackrel{\u02c6}{\mathit{\beta}}\in (0,1)$, $\frac{1}{{\mathit{\beta}}_{\mathit{\epsilon}}}(log{u}_{\mathit{\epsilon}}-\mathbb{E}[log{u}_{\mathit{\epsilon}}])$ converges in distribution as a random field to a solution of Edwards-Wilkinson equation.

In this paper, we consider a stochastic heat equation ${u}_{\mathit{\epsilon}}$ with a general initial condition ${u}_{0}$ and its transformation $F({u}_{\mathit{\epsilon}})$ for *F* in a class of functions $\mathfrak{F}$, which contains $F(x)={x}^{p}$ ($0<p\le 1$) and $F(x)=logx$. Then, we prove that $\frac{1}{{\mathit{\beta}}_{\mathit{\epsilon}}}(F({u}_{\mathit{\epsilon}}(t,\cdot ))-\mathbb{E}[F({u}_{\mathit{\epsilon}}(t,\cdot ))])$ converges in distribution as a random field to a centered Gaussian field jointly in finitely many $F\in \mathfrak{F}$, *t*, and ${u}_{0}$. In particular, we show the fluctuations of solutions of stochastic heat equations and KPZ equations jointly converge to solutions of SPDEs which depend on ${u}_{0}$.

Our main tools are Itô’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.

## Funding Statement

The work of S. Nakajima is supported by SNSF grant 176918. M. Nakashima is supported by JSPS KAKENHI Grant Numbers JP18H01123, JP18K13423.

## Acknowledgments

The authors would like to thank anonymous referees for many useful comments and suggestions.

## Citation

Shuta Nakajima. Makoto Nakashima. "Fluctuations of two-dimensional stochastic heat equation and KPZ equation in subcritical regime for general initial conditions." Electron. J. Probab. 28 1 - 38, 2023. https://doi.org/10.1214/22-EJP885

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