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We study the equilibrium fluctuations of an interacting particle system evolving on the discrete ring with N ∈ N points, denoted by T N , and with three species of particles that we name A, B and C, but such that at each site there is only one particle. We prove that proper choices of density fluctuation fields (that match those from nonlinear fluctuating hydrodynamics theory) associated to the (two) conserved quantities converge, in the limit N → ∞ , to a system of stochastic partial differential equations, that can either be the Ornstein–Uhlenbeck equation or the Stochastic Burgers equation. To understand the cross interaction between the two conserved quantities, we derive a general version of the Riemann–Lebesgue lemma which is of independent interest. [ABSTRACT FROM AUTHOR]

For the KPZ equation on a torus with a 1 + 1 spacetime white noise, it was shown in Dunlap et al. (Commun Pure Appl Math, 2023, https://doi.org/10.1002/cpa.22110) and Gu and Komorowski (Ann Inst H Poincare Prob Stat, 2021, arXiv:2104.13540v2) that the height function satisfies a central limit theorem, and the variance can be written as the expectation of an exponential functional of Brownian bridges. In this paper, we consider another physically relevant quantity, the winding number of the directed polymer on a cylinder, or equivalently, the displacement of the directed polymer endpoint in a spatially periodic random environment. It was shown in Gu and Komorowski (SIAM J Math Anal, arXiv:2207.14091) that the polymer endpoint satisfies a central limit theorem on diffusive scales. The main result of this paper is an explicit expression of the effective diffusivity, in terms of the expectation of another exponential functional of Brownian bridges. Our argument is based on a combination of tools from Malliavin calculus, homogenization, and diffusion in distribution-valued random environments. [ABSTRACT FROM AUTHOR]

The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation which is derived from various microscopic models, and to establish a robust way to derive the KPZ equation is a fundamental problem both in mathematics and in physics. As a microscopic model, we consider multi-species interacting diffusion processes, whose dynamics is driven by a nonlinear potential which satisfies some regularity conditions. In particular, we study asymptotic behavior of fluctuation fields associated with the processes in the high temperature regime under equilibrium. As a main result, we show that when the characteristic speed of each species is the same, the family of the fluctuation fields seen in moving frame with this speed converges to the coupled KPZ equations. Our approach is based on a Taylor expansion argument which extracts the harmonic potential as a main part. This argument works without assuming a specific form of the potential and thereby the coupled KPZ equations are derived in a robust way. [ABSTRACT FROM AUTHOR]

The Kardar–Parisi–Zhang (KPZ) equation describes a wide range of growth-like phenomena, with applications in physics, chemistry and biology. There are three central questions in the study of KPZ growth: the determination of height probability distributions; the search for ever more precise universal growth exponents; and the apparent absence of a fluctuation–dissipation theorem (FDT) for spatial dimension d > 1 . Notably, these questions were answered exactly only for 1 + 1 dimensions. In this work, we propose a new FDT valid for the KPZ problem in d + 1 dimensions. This is achieved by rearranging terms and identifying a new correlated noise which we argue to be characterized by a fractal dimension d n . We present relations between the KPZ exponents and two emergent fractal dimensions, namely d f , of the rough interface, and d n . Also, we simulate KPZ growth to obtain values for transient versions of the roughness exponent α , the surface fractal dimension d f and, through our relations, the noise fractal dimension d n . Our results indicate that KPZ may have at least two fractal dimensions and that, within this proposal, an FDT is restored. Finally, we provide new insights into the old question about the upper critical dimension of the KPZ universality class. [ABSTRACT FROM AUTHOR]

We study the limit of a local average of the KPZ equation in dimension d = 2 with general initial data in the subcritical regime. Our result shows that a proper spatial averaging of the KPZ equation converges in distribution to the sum of the solution to a deterministic KPZ equation and a Gaussian random variable that depends solely on the scale of averaging. This shows a unique mesoscopic averaging phenomenon that is only present in dimension two. Our work is inspired by the recent findings by Chatterjee (Ann l’Institut Henri Poincare (B) Probabilites et statistiques 59(2):774–794, 2023). [ABSTRACT FROM AUTHOR]

We consider directed polymers in random environment in the critical dimension d = 2 , focusing on the intermediate disorder regime when the model undergoes a phase transition. We prove that, at criticality, the diffusively rescaled random field of partition functions has a unique scaling limit: a universal process of random measures on R 2 with logarithmic correlations, which we call the Critical 2d Stochastic Heat Flow. It is the natural candidate for the long sought solution of the critical 2d Stochastic Heat Equation with multiplicative space-time white noise. [ABSTRACT FROM AUTHOR]

We study the KPZ equation with a 1+1-dimensional spacetime white noise, started at equilibrium, and give a different proof of the main result of Balázs, Quastel, and Seppäläinen [J. Amer. Math. Soc. 24 (2011), pp. 683–708], i.e., the variance of the solution at time t is of order t^{2/3}. Instead of using a discrete approximation through the exclusion process and the second class particle, we utilize the connection to directed polymers in random environment. Along the way, we show the annealed density of the stationary continuum directed polymer equals to the two-point covariance function of the stationary stochastic Burgers equation, confirming the physics prediction of Maes and Thiery [J. Stat. Phys. 168 (2017), pp. 937–963]. [ABSTRACT FROM AUTHOR]

Sharp large deviation estimates for stochastic differential equations with small noise, based on minimizing the Freidlin–Wentzell action functional under appropriate boundary conditions, can be obtained by integrating certain matrix Riccati differential equations along the large deviation minimizers or instantons, either forward or backward in time. Previous works in this direction often rely on the existence of isolated minimizers with positive definite second variation. By adopting techniques from field theory and explicitly evaluating the large deviation prefactors as functional determinant ratios using Forman’s theorem, we extend the approach to general systems where degenerate submanifolds of minimizers exist. The key technique for this is a boundary-type regularization of the second variation operator. This extension is particularly relevant if the system possesses continuous symmetries that are broken by the instantons. We find that removing the vanishing eigenvalues associated with the zero modes is possible within the Riccati formulation and amounts to modifying the initial or final conditions and evaluation of the Riccati matrices. We apply our results in multiple examples including a dynamical phase transition for the average surface height in short-time large deviations of the one-dimensional Kardar–Parisi–Zhang equation with flat initial profile. [ABSTRACT FROM AUTHOR]

We prove a frequency-independent bound on trigonometric functions of a class of singular Gaussian random fields, which arise naturally from weak universality problems for singular stochastic PDEs. This enables us to reduce the regularity assumption on the nonlinearity of the microscopic models (for pathwise convergence) in KPZ and dynamical Φ 3 4 in the previous works of Hairer-Xu and Furlan-Gubinelli to heuristically optimal thresholds required by PDE structures. [ABSTRACT FROM AUTHOR]

We consider equilibrium fluctuations for a class of totally asymmetric zero-range type interacting particle systems, which is particularly related to q -totally asymmetric simple exclusion processes (q -TASEPs). As a main result, we show that density fluctuation of our process converges to the stationary energy solution of the stochastic Burgers equation. Our proof is based on the second-order Boltzmann–Gibbs principle and a Taylor expansion argument. [ABSTRACT FROM AUTHOR]

We have analysed the research findings on the universality class and discussed the connection between the Kardar-Parisi-Zhang (KPZ) universality class and the ballistic deposition model in microscopic rules. In one dimension and 1+1 dimensions deviations are not important in the presence of noise. At the same time, they are very relevant for higher dimensions or deterministic evolution. Mostly, in the analyses a correction scale higher than 1280 has not been studied yet. Therefore, the growth of the interface for finite system size β ≥ 0.30 value predicted by the KPZ universality class is still predominant. Also, values of α ≥ 0.40, β ≥ 0.30, and z ≥ 1.16 obtained from literature are consistent with the expected KPZ values of α = 1 / 2 , β = 1 / 3 , and z = 3 / 2 . A connection between the ballistic deposition and the KPZ equation through the limiting procedure and by applying the perturbation method was also presented.

We study the long-time behavior of Kardar–Parisi–Zhang (KPZ)-like equations: ∂ t h (t , x) = Δ x h (t , x) + | ∇ x h (t , x) | 2 + η (t , x) , h (0 , x) = h 0 (x) , (t , x) ∈ (0 , ∞) × d , on the d -dimensional torus d driven by an ergodic noise η (e.g., space-time white in d = 1). The analysis builds on infinite-dimensional extensions of similar results for positive random matrices. We establish a one force, one solution principle and derive almost sure synchronization with exponential deterministic speed in appropriate Hölder spaces. [ABSTRACT FROM AUTHOR]

The Kardar–Parisi-Zhang (KPZ) equation is examined using the recently published leapfrog–hopscotch (LH) method as well as the most standard forward time centered space (FTCS) scheme and the Heun method. The methods are verified by reproducing an analytical solution. The performance of each method is then compared by calculating the average and the maximum differences among the results and displaying the runtimes. Numerical tests show that due to the special symmetry in the time–space discretisation, the new LH method clearly outperforms the other two methods. In addition, we discuss the effect of different parameters on the solutions. [ABSTRACT FROM AUTHOR]

Applications and modeling of various phenomena in all areas of scientific research require finding numerical solutions for differential, partial differential, integral, or integro-differential equations. In addition to proving theoretical convergence and giving error estimates, stability of numerical methods for such operator equations is a fundamental property that it is necessary for the method to produce a valid solution. This Special Issue focuses on new theoretical and numerical studies concerning the techniques used for proving stability or instability of numerical schemes, which extend or improve known results. It also includes applications to non-linear physical, chemical, and engineering systems, arising in dynamics of waves, diffusion, or transport problems. [ABSTRACT FROM AUTHOR]

The Kardar-Parisi-Zhang (KPZ) equation has been connected to a large number of important stochastic processes in physics, chemistry and growth phenomena, ranging from classical to quantum physics. The central quest in this field is the search for ever more precise universal growth exponents. Notably, exact growth exponents are only known for 1+1 dimensions. In this work, we present physical and geometric analytical methods that directly associate these exponents to the fractal dimension of the rough interface. Based on this, we determine the growth exponents for the 2+1 dimensions, which are in agreement with the results of thin films experiments and precise simulations. We also make a first step towards a solution in d+1 dimensions, where our results suggest the inexistence of an upper critical dimension.

We introduce a stochastic eight-vertex model motivated by the eight-vertex model in statistical mechanics. We study its invariant measures under a certain balance condition for the creation and annihilation probabilities of particles. We also discuss its scaling limits, that is, the hydrodynamic limit, and the linear and nonlinear fluctuations rather formally. We especially obtain a new type of KPZ-Burgers equation in the nonlinear limit. [ABSTRACT FROM AUTHOR]

The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the self-similar ansatz is analyzed. As a new feature additional analytic terms are added. From the mathematical point of view, these can be considered as various noise distribution functions. Six different cases were investigated among others Gaussian, Lorentzian, white or even pink noise. Analytic solutions are evaluated and analyzed for all cases. All results are expressible with various special functions like Kummer, Heun, Whittaker or error functions showing a very rich mathematical structure with some common general characteristics.

In this paper, we study several aspects related with solutions of nonlocal problems whose prototype is u t = ∫ ℝ N J (x − y) (u (y , t) − u (x , t)) 𝒢 (u (y , t) − u (x , t)) d y in  Ω × (0 , T) , u (x , 0) = u 0 (x) in  Ω , where we take, as the most important instance, 𝒢 (s) ∼ 1 + μ 2 s 1 + μ 2 s 2 with μ ∈ ℝ as well as u 0 ∈ L 1 (Ω) , J is a smooth symmetric function with compact support and Ω is either a bounded smooth subset of ℝ N , with nonlocal Dirichlet boundary condition, or ℝ N itself. The results deal with existence, uniqueness, comparison principle and asymptotic behavior. Moreover, we prove that if the kernel is rescaled in a suitable way, the unique solution of the above problem converges to a solution of the deterministic Kardar–Parisi–Zhang equation. [ABSTRACT FROM AUTHOR]

We consider the stationary O'Connell–Yor model of semi-discrete directed polymers in a Brownian environment in the intermediate disorder regime and show convergence of the increments of the log-partition function to the energy solutions of the stochastic Burgers equation. The proof does not rely on the Cole–Hopf transform and avoids the use of spectral gap estimates for the discrete model. The key technical argument is a second-order Boltzmann–Gibbs principle. [ABSTRACT FROM AUTHOR]

The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the self-similar ansatz is analyzed. As a new feature additional analytic terms are added. From the mathematical point of view, these can be considered as various noise distribution functions. Six different cases were investigated among others Gaussian, Lorentzian, white or even pink noise. Analytic solutions are evaluated and analyzed for all cases. All results are expressible with various special functions like Kummer, Heun, Whittaker or error functions showing a very rich mathematical structure with some common general characteristics. [ABSTRACT FROM AUTHOR]

We consider the stochastic higher spin six vertex (SHS6V) model introduced by Corwin and Petrov (Commun. Math. Phys., 343(2), 651–700 2016) with general integer spin parameters I, J. Starting from near stationary initial condition, we prove that the SHS6V model converges to the Kardar-Parisi-Zhang (KPZ) equation under weakly asymmetric scaling. This generalizes the result in Corwin et al. (2018, Theorem 1.1) from I = J = 1 to general I, J. [ABSTRACT FROM AUTHOR]

We consider the Brownian directed polymer in Poissonian environment in dimension 1+1, under the so-called intermediate disorder regime (Alberts et al., 2014), which is a crossover regime between the strong and weak disorder regions. We show that, under a diffusive scaling involving different parameters of the system, the renormalized point-to-point partition function of the polymer converges in law to the solution of the stochastic heat equation with Gaussian multiplicative noise. The Poissonian environment provides a natural setting and strong tools, such as the Wiener–Itô chaos expansion (Last and Penrose, 2017), which, applied to the partition function, is the basic ingredient of the proof. [ABSTRACT FROM AUTHOR]

The one-dimensional Kardar-Parisi-Zhang dynamic interface growth equation with the self-similar ansatz is analyzed. As a new feature additional analytic terms are added. From the mathematical point of view, these can be considered as various noise distribution functions. Six different cases were investigated among others Gaussian, Lorentzian, white or even pink noise. Analytic solutions are evaluated and analyzed for all cases. All results are expressible with various special functions like Kummer, Heun, Whittaker or error functions showing a very rich mathematical structure with some common general characteristics. [ABSTRACT FROM AUTHOR]

We study in the present article the Kardar-Parisi-Zhang (KPZ) equation ∂th(t,x)=νΔh(t,x)+λ|∇h(t,x)|2+Dη(t,x),(t,x)∈R+×Rdin d≥3 dimensions in the perturbative regime, i.e. for λ>0 small enough and a smooth, bounded, integrable initial condition h0=h(t=0,·). The forcing term η in the right-hand side is a regularized space-time white noise. The exponential of h—its so-called Cole-Hopf transform—is known to satisfy a linear PDE with multiplicative noise. We prove a large-scale diffusive limit for the solution, in particular a time-integrated heat-kernel behavior for the covariance in a parabolic scaling. The proof is based on a rigorous implementation of K. Wilson’s renormalization group scheme. A double cluster/momentum-decoupling expansion allows for perturbative estimates of the bare resolvent of the Cole-Hopf linear PDE in the small-field region where the noise is not too large, following the broad lines of Iagolnitzer and Magnen (Commun Math Phys 162(1):85-121, 1994). Standard large deviation estimates for η make it possible to extend the above estimates to the large-field region. Finally, we show, by resumming all the by-products of the expansion, that the solution h may be written in the large-scale limit (after a suitable Galilei transformation) as a small perturbation of the solution of the underlying linear Edwards-Wilkinson model (λ=0) with renormalized coefficients νeff=ν+O(λ2),Deff=D+O(λ2). [ABSTRACT FROM AUTHOR]

We show that a generalized Asymmetric Exclusion Process called ASEP(q, j ) introduced in (Probab. Theory Related Fields 166 (2016) 887-933). converges to the Cole-Hopf solution to the KPZ equation under weak asymmetry scaling. [ABSTRACT FROM AUTHOR]

In this paper we consider a probability distribution on plane partitions, which arises as a one-parameter generalization of the standard qvolume measure. This generalization is closely related to the classical multivariate Hall-Littlewood polynomials, and it was first introduced by Vuleti-c in (Trans. Am. Math. Soc. 361 (2009) 2789-2804). We prove that as the plane partitions become large (q goes to 1, while the Hall-Littlewood parameter t is fixed), the scaled bottom slice of the random plane partition converges to a deterministic limit shape, and that one-point fluctuations around the limit shape are asymptotically given by the GUE Tracy-Widom distribution. On the other hand, if t simultaneously converges to its own critical value of 1, the fluctuations instead converge to the one-dimensional Kardar-Parisi-Zhang (KPZ) equation with the so-called narrow wedge initial data. The algebraic part of our arguments is closely related to the formalism of Macdonald processes (Probab. Theory Relat. Fields 158 (1) (2014) 225-400). The analytic part consists of detailed asymptotic analysis of the arising Fredholm determinants. [ABSTRACT FROM AUTHOR]

In this paper, we consider the approximating KPZ equation introduced by Funaki and Quastel (2015), which is suitable for studying invariant measures. They showed that the stationary solution of the approximating equation converges to the Cole–Hopf solution of the KPZ equation with extra term 1 24 t . On the other hand, Gubinelli and Perkowski (2017) gave a pathwise meaning to the KPZ equation as an application of the paracontrolled calculus. We show that Funaki and Quastel’s result is extended to nonstationary solutions by using the paracontrolled calculus. [ABSTRACT FROM AUTHOR]

This paper presents a method to obtain semi-analytical solutions to second order semilinear IBVPs where the standard Adomian decomposition may fail to yield nontrivial solutions or fail to produce the correct partial solutions. In contrast to the standard Adomian decomposition method, the proposed solution method is distinguished by simultaneous inversion of the linear differential operators using eigenfunctions expansion representations. The proposed method is applied to several examples of initial boundary value problems — a linear Advection–Diffusion problem, a Burgers equation and the deterministic Kardar–Parisi–Zhang (KPZ) equation. It is shown that the dependence of the series solution on the truncation order ( N ), the number of eigenfunctions ( M ) and the diffusion coefficient is rather complex. [ABSTRACT FROM AUTHOR]

We consider one-dimensional exclusion processes with long jumps given by a transition probability of the form p n ( ⋅ ) = s ( ⋅ ) + γ n a ( ⋅ ) , such that its symmetric part s ( ⋅ ) is irreducible with finite variance and its antisymmetric part is absolutely bounded by s ( ⋅ ) . We prove that under diffusive time scaling and strength of asymmetry n γ n → n → ∞ b ≠ 0 , the equilibrium density fluctuations are given by the unique energy solution of the stochastic Burgers equation. [ABSTRACT FROM AUTHOR]

This paper concerns the multi-component coupled Kardar–Parisi–Zhang (KPZ) equation and its two types of approximations. One approximation is obtained as a simple replacement of the noise term by a smeared noise with a proper renormalization, while the other one introduced in [6] is suitable for studying the invariant measures. By applying the paracontrolled calculus introduced by Gubinelli et al. [8,9] , we show that two approximations have the common limit under the properly adjusted choice of renormalization factors for each of these approximations. In particular, if the coupling constants of the nonlinear term of the coupled KPZ equation satisfy the so-called “trilinear” condition, the renormalization factors can be taken the same in two approximations and the difference of the limits of two approximations are explicitly computed. Moreover, under the trilinear condition, the Wiener measure twisted by the diffusion matrix becomes stationary for the limit and we show that the solution of the limit equation exists globally in time when the initial value is sampled from the stationary measure. This is shown for the associated tilt process. Combined with the strong Feller property shown by Hairer and Mattingly [12] , this result can be extended for all initial values. [ABSTRACT FROM AUTHOR]

We use Renormalization Group to prove local well posedness for a generalized KPZ equation introduced by H. Spohn in the context of stochastic hydrodynamics. The equation requires the addition of counter terms diverging with a cutoff $$\epsilon $$ as $$\epsilon ^{-1}$$ and $$\log \epsilon ^{-1}$$ . [ABSTRACT FROM AUTHOR]

We consider the transition probabilities for random walks in $$1+1$$ dimensional space-time random environments (RWRE). For critically tuned weak disorder we prove a sharp large deviation result: after appropriate rescaling, the transition probabilities for the RWRE evaluated in the large deviation regime, converge to the solution to the stochastic heat equation (SHE) with multiplicative noise (the logarithm of which is the KPZ equation). We apply this to the exactly solvable Beta RWRE and additionally present a formal derivation of the convergence of certain moment formulas for that model to those for the SHE. [ABSTRACT FROM AUTHOR]

We study the BS model, which is a one-dimensional lattice field theory taking real values. Its dynamics is governed by coupled differential equations plus random nearest neighbor exchanges. The BS model has two locally conserved fields. The peak structure of their steady state space-time correlations is determined through numerical simulations and compared with nonlinear fluctuating hydrodynamics, which predicts a traveling peak with KPZ scaling function and a standing peak with a scaling function given by the maximally asymmetric Lévy distribution with parameter $$\alpha = 5/3$$ . As a by-product, we completely classify the universality classes for two coupled stochastic Burgers equations with arbitrary coupling coefficients. [ABSTRACT FROM AUTHOR]

The one dimensional Kardar-Parisi-Zhang universality class is believed to describe many types of evolving interfaces which have the same characteristic scaling exponents. These exponents lead to a natural renormalization/rescaling on the space of such evolving interfaces. We introduce and describe the renormalization fixed point of the Kardar-Parisi-Zhang universality class in terms of a random nonlinear semigroup with stationary independent increments, and via a variational formula. Furthermore, we compute a plausible formula the exact transition probabilities using replica Bethe ansatz. The semigroup is constructed from the Airy sheet, a four parameter space-time field which is the Airy $$_2$$ process in each of its two spatial coordinates. Minimizing paths through this field describe the renormalization group fixed point of directed polymers in a random potential. At present, the results we provide do not have mathematically rigorous proofs, and they should at most be considered proposals. [ABSTRACT FROM AUTHOR]

We compute the non-universal constants in the KPZ equation in one dimension, in terms of the thermodynamical quantities associated to the underlying microscopic dynamics. In particular, we derive the second-order Einstein relation $$\lambda = \frac{a}{2}\frac{d^2}{d\rho ^2} \chi (\rho ) D(\rho )$$ for the transport coefficient $$\lambda $$ of the KPZ equation, in terms of the conserved quantity $$\rho $$ , the diffusion coefficient $$D$$ , the strength of the asymmetry $$a$$ and the static compressibility of the system $$\chi $$ . [ABSTRACT FROM AUTHOR]