Phenomena of non-equilibrium growth processes are ubiquitous in nature. Many of these processes are believed to exhibit remarkably similar growth dynamics, which can be regarded as an interface that evolves with time, changing its roughness while being subjected to random noise. In the original 1986 paper of M. Kardar, G. Parisi and Y.-C. Zhang [2], the authors predicted the dynamic of these processes, and indicated that the evolution can be described by the solution of a stochastic PDE, which is called the Kardar-Parisi-Zhang (KPZ) equation. On the other hand, plays an important role in statistical mechanics. Roughly speaking, through the Cole–Hopf transform , can be regarded as the free energy of the continuous directed polymer model, which describes the behaviour of a hydrophilic polymer chain wafting in a disordered environment that contains randomly placed hydrophobic molecules as impurities. Here plays the role of the partition function of this model, and solves a stochastic PDE, which is called

the stochastic heat equation (SHE).

 

In this talk, we will focus on the SHE and KPZ equation when spatial dimension . In this case, both of the SHE and KPZ equation are quite sensitive to , where describes the strength of the white noise in these equations. Consequently, it is crucial to investigate the behavior of the SHE and KPZ equation for all . In a recent paper of F. Comets, C. Cosco, and C. Mukherjee [1], the authors proved the limiting fluctuation of under a restriction on . Motivated by this result, in our recent work [3], we considered both and , and established their limiting fluctuations in the entire -regime (i.e., ). Here is a critical value associated with the KPZ equation. 

 

 

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