Fluctuations of two-dimensional stochastic heat equation and KPZ equation in subcritical regime for general initial conditions. (English) Zbl 1515.60334

Nakajima, Shuta; Nakashima, Makoto

Electron. J. Probab. 28, Paper No. 1, 38 p. (2023).

Summary: The Kardar-Parisi-Zhang equation (KPZ equation) is solved via Cole-Hopf transformation \(h=\log u\), where \(u\) is the solution of the multiplicative stochastic heat equation(SHE). In [S. Chatterjee and A. Dunlap, Ann. Probab. 48, No. 2, 1014–1055 (2020; Zbl 1434.60148); F. Caravenna et al., Ann. Probab. 48, No. 3, 1086–1127 (2020; Zbl 1444.60061); Y. Gu, Stoch. Partial Differ. Equ., Anal. Comput. 8, No. 1, 150–185 (2020; Zbl 1431.35257)], they consider the solution of two dimensional KPZ equation via the solution \(u_{\varepsilon}\) of SHE with the flat initial condition and with noise which is mollified in space on scale in \(\varepsilon\) and its strength is weakened as \(\beta_{\varepsilon}=\hat{\beta}\sqrt{\frac{2\pi}{-\log \varepsilon}}\), 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 as a random field to a solution of Edwards-Wilkinson equation.
In this paper, we consider a stochastic heat equation \(u_{\varepsilon}\) with a 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 \(\frac{1}{\beta_{\varepsilon}}(F(u_{\varepsilon}(t,\cdot))-\mathbb{E}[F(u_{\varepsilon}(t,\cdot))])\) 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 [C. Cosco et al., Stochastic Processes Appl. 151, 127–173 (2022; Zbl 1493.60149)]. To this end, we also prove a local limit theorem for the partition function of intermediate disorder \(2d\) directed polymers.

MSC:

60K37	Processes in random environments
60F05	Central limit and other weak theorems
60G44	Martingales with continuous parameter
82D60	Statistical mechanics of polymers

Keywords:

Edwards-Wilkinson equation; KPZ equation; local limit theorem for polymers; stochastic calculus; stochastic heat equation

Citations:

Zbl 1434.60148; Zbl 1444.60061; Zbl 1431.35257; Zbl 1493.60149

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