We trace the evolution of theory stochastic partial differential equations from foundation to its development, until recent solution long-standing problems on well-posedness KPZ equation and quantization in dimension three.

A brief review is made of the birth and evolution “nonequilibrium potential” (NEP) concept. As if providing a landscape for qualitative reasoning were not helpful enough, NEP adds quantitative dimension to theory differential equations provides global Lyapunov function deterministic dynamics. Here we illustrate usefulness draw results on stochastic thermodynamics: Jarzynski equality in Wilson–C...

Abstract We investigate the thermodynamic uncertainty relation for $$(1+1)$$ ( 1 + ) dimensional Kardar–Parisi–Zhang (KPZ) equation on a finite spatial interval. In particular, we extend results small coupling strengths obtained previou...

We determine the stationary two-point correlation function of the onedimensional KPZ equation through the scaling limit of a solvable microscopic model, the polynuclear growth model. The equivalence to a directed polymer problem with specific boundary conditions allows one to express the corresponding scaling function in terms of the solution to a Riemann-Hilbert problem related to the Painlevé...

Spatiotemporal chaos (STC) exhibited by the Kuramoto-Sivashinsky (KS) equation is investigated analytically and numerically. An eeective stochastic equation belonging to the KPZ universality class is constructed by incorporating the chaotic dynamics of the small KS system in a coarse-graining procedure. The bare parameters of the eeective theory are computed approximately. Stability of the syst...

In order to estimate roughness exponents of interface growth models, we propose the calculation of effective exponents from the roughness fluctuation σ in the steady state. We compare the finite-size behavior of these exponents and the ones calculated from the average roughness 〈w2〉 for two models in the 2 + 1-dimensional KardarParisi-Zhang (KPZ) class and for a model in the 1 + 1-dimensional V...

The results of Amir–Corwin–Quastel, Calabrese–Le Doussal–Rosso, Dotsenko, and Sasamoto–Spohn imply that the one-point distribution of the solution of the KPZ equation with the narrow wedge initial condition coincides with that for a multiplicative statistics of the Airy determinantal random point process. Taking Taylor coefficients of the two sides yields moment identities. We provide a simple ...

We study a moving-boundary model of nonconserved interface growth that implements the interplay between diffusive matter transport and aggregation kinetics at the interface. Conspicuous examples are found in thin-film production by chemical vapor deposition and electrochemical deposition. The model also incorporates noise terms that account for fluctuations in the diffusive and attachment proce...

Results of experiments on the dynamics and kinetic roughening of one-dimensional slow-combustion fronts in three grades of paper are reported. Extensive averaging of the data allows a detailed analysis of the spatial and temporal development of the interface fluctuations. The asymptotic scaling properties, on long length and time scales, are well described by the Kardar-Parisi-Zhang (KPZ) equat...