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

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 ${\mathit{\beta }}_{\mathit{\epsilon }}=\hat{\mathit{\beta }}\sqrt{\frac{2\mathrm{\pi }}{-log\mathit{\epsilon }}}$, and they prove that when $\hat{\mathit{\beta }}\in (0,1)$, $\frac{1}{{\mathit{\beta }}_{\mathit{\epsilon }}}(log{u}_{\mathit{\epsilon }}-\mathbb{E}[log{u}_{\mathit{\epsilon }}])$ 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.