Hopf-Cole Solution Research Articles

Directed mean curvature flow in noisy environment

AbstractWe consider the directed mean curvature flow on the plane in a weak Gaussian random environment. We prove that, when started from a sufficiently flat initial condition, a rescaled and recentred solution converges to the Cole–Hopf solution of the KPZ equation. This result follows from the analysis of a more general system of nonlinear SPDEs driven by inhomogeneous noises, using the theory of regularity structures. However, due to inhomogeneity of the noise, the “black box” result developed in the series of works cannot be applied directly and requires significant extension to infinite‐dimensional regularity structures. Analysis of this general system of SPDEs gives two more interesting results. First, we prove that the solution of the quenched KPZ equation with a very strong force also converges to the Cole–Hopf solution of the KPZ equation. Second, we show that a properly rescaled and renormalised quenched Edwards–Wilkinson model in any dimension converges to the stochastic heat equation.

Characterization of H-Brownian Gibbsian line ensembles

In this paper we show that an $H$-Brownian Gibbsian line ensemble is completely characterized by the finite-dimensional marginals of its lowest indexed curve for a large class of interaction Hamiltonians $H$. A particular consequence of our result is that the KPZ line ensemble is the unique line ensemble that satisfies the $H_1$-Brownian Gibbs property with $H_1(x) = e^x$ and whose lowest indexed curve is equal to the Hopf-Cole solution to the narrow wedge KPZ equation.

Exact lower-tail large deviations of the KPZ equation

Consider the Hopf–Cole solution h(t,x) of the Kandar–Parisi–Zhang (KPZ) equation with the narrow wedge initial condition. Regarding t→∞ as a scaling parameter, we provide the first rigorous proof of the large deviation principle (LDP) for the lower tail of h(2t,0)+t 12, with speed t2 and an explicit rate function Φ −(z). This result confirms existing physics predictions made by Corwin (2011); Sasorov, Meerson, and Prolhac (2017); and Krajenbrink, Le Doussal, and Prolhac (2018). Our analysis utilizes a formula from Borodin and Gorin (2016) to convert the LDP for the KPZ equation to calculating an exponential moment of the Airy point process (PP). To estimate this exponential moment, we invoke the stochastic Airy operator (SAO) and use the Riccati transform, comparison techniques, and certain variational characterizations of the relevant functional.

Burgers equation and convexification method: Propagation of wave packets and narrowband noise.

The Burgers equation is a standard model for the propagation of progressive finite amplitude waves in a lossy medium. This paper is mainly devoted to the one-dimensional Burgers equation and its solution deduced from a convexification method. Its starting point is the Hopf-Cole solution in the inviscid limit, then the Legendre-Fenchel transform and the convex envelope construction are employed to obtain the single-value waveform. A fractional step method is used to numerically solve the Burgers equation in a segregated manner (nonlinearity and dissipation). Numerical results are obtained for nonlinear propagation of a wave packet with a Gaussian envelope and a narrowband noise (deterministic and non-deterministic signals). The convexification method provides a tool to solve the problems of nonlinear propagation, especially in the case of noise propagation with the formation and interaction (coalescence) of a large number of discontinuities (shocks).

Central limit theorems for spatial averages of the stochastic heat equation via Malliavin–Stein’s method

Suppose that \(\{u(t, x)\}_{t >0, x \in {\mathbb {R}}^d}\) is the solution to a d-dimensional stochastic heat equation driven by a Gaussian noise that is white in time and has a spatially homogeneous covariance that satisfies Dalang’s condition. The purpose of this paper is to establish quantitative central limit theorems for spatial averages of the form \(N^{-d} \int _{[0,N]^d} g(u(t,x))\, \mathrm {d}x\), as \(N\rightarrow \infty \), where g is a Lipschitz-continuous function or belongs to a class of locally-Lipschitz functions, using a combination of the Malliavin calculus and Stein’s method for normal approximations. Our results include a central limit theorem for the Hopf–Cole solution to KPZ equation. We also establish a functional central limit theorem for these spatial averages.

Derivation of the stochastic Burgers equation with Dirichlet boundary conditions from the WASEP

We consider the weakly asymmetric simple exclusion process on the discrete space {1,⋯,n-1}(n∈ℕ), in contact with stochastic reservoirs, both with density ρ∈(0,1) at the extremity points, and starting from the invariant state, namely the Bernoulli product measure of parameter ρ. Under time diffusive scaling tn 2 and for ρ=1 2, when the asymmetry parameter is taken of order 1/n, we prove that the density fluctuations at stationarity are macroscopically governed by the energy solution of the stochastic Burgers equation with Dirichlet boundary conditions, which is shown to be unique and to exhibit different boundary behavior than the Cole–Hopf solution.

KPZ equation tails for general initial data

We consider the upper and lower tail probabilities for the centered (by time$/24$) and scaled (according to KPZ time$^{1/3}$ scaling) one-point distribution of the Cole-Hopf solution of the KPZ equation when started with initial data drawn from a very general class. For the lower tail, we prove an upper bound which demonstrates a crossover from super-exponential decay with exponent $3$ in the shallow tail to an exponent $5/2$ in the deep tail. For the upper tail, we prove super-exponential decay bounds with exponent $3/2$ at all depths in the tail.

Borodin\u2013P\xe9ch\xe9 Fluctuations of the Free Energy in Directed Random Polymer Models

We consider two directed polymer models in the Kardar–Parisi–Zhang (KPZ) universality class: the O’Connell–Yor semi-discrete directed polymer with boundary sources and the continuum directed random polymer with (m, n)-spiked boundary perturbations. The free energy of the continuum polymer is the Hopf–Cole solution of the KPZ equation with the corresponding (m, n)-spiked initial condition. This new initial condition is constructed using two semi-discrete polymer models with independent bulk randomness and coupled boundary sources. We prove that the limiting fluctuations of the free energies rescaled by the 1 / 3rd power of time in both polymer models converge to the Borodin–Péché-type deformations of the GUE Tracy–Widom distribution.

Local Solution to the Multi-layer KPZ Equation

In this article we prove local well-posedness of the system of equations $\partial_t h_{i}= \sum_{j=1}^{i}\partial_x^2 h_{i}+ (\partial_x h_{i})^2 + \xi $ on the circle where $1\leq i\leq N$ and $\xi$ is a space-time white noise. We attempt to generalize the renormalization procedure which gives the Hopf-Cole solution for the single layer equation and our $h_1$ (solution to the first layer) coincides with this solution. However, we observe that cancellation of logarithmic divergences that occurs at the first layer does not hold at higher layers and develop explicit combinatorial formulae for them.

A brief introduction to KPZ equation and KPZ universality

In this survey, we shall first introduce the physical background of KPZ universality class.The Eden model, sticky block model and KPZ equation will be mentioned.Then we shall focus on the Cole-Hopf solution to 1-dimensional KPZ equation and present some discrete models like the corner growth model and directed polymer model,which could converge to the KPZ equation in some sense.

The KPZ Equation and Moments of Random Matrices

The logarithm of the diagonal matrix element of a high power of a random matrix converges to the Cole-Hopf solution of the Kardar-Parisi-Zhang equation in the sense of one-point distributions.

The KPZ Limit of ASEP with Boundary

It was recently proved in [Corwin-Shen, 2016] that under weak asymmetry scaling, the height functions for open ASEP on the half-line and on a bounded interval converge to the Hopf-Cole solution of the KPZ equation with Neumann boundary conditions. In their assumptions [Corwin-Shen, 2016] chose positive values for the Neumann boundary conditions, and they assumed initial data which is close to stationarity. By developing more extensive heat-kernel estimates, we extend their results to negative values of the Neumann boundary parameters, and we also show how to generalize their results to narrow-wedge initial data (which is very far from stationarity). As a corollary via [Barraquand-Borodin-Corwin-Wheeler, 2017], we obtain the Laplace transform of the one-point distribution for half-line KPZ, and use this to prove $t^{1/3}$-scale GOE Tracy-Widom long-time fluctuations.

$\operatorname{ASEP}(q,j)$ converges to the KPZ equation

Nous montrons qu’une généralisation du processus d’exclusion asymétrique appelée $\operatorname{ASEP}(q,j)$, introduite dans (Probab. Theory Related Fields 166 (2016) 887–933), converge sous faible asymétrie vers la solution de l’équation KPZ au sens de Cole–Hopf.

A central limit theorem for the KPZ equation

We consider the KPZ equation in one space dimension driven by a stationary centred space-time random field, which is sufficiently integrable and mixing, but not necessarily Gaussian. We show that, in the weakly asymmetric regime, the solution to this equation considered at a suitable large scale and in a suitable reference frame converges to the Hopf-Cole solution to the KPZ equation driven by space-time Gaussian white noise. While the limiting process depends only on the integrated variance of the driving field, the diverging constants appearing in the definition of the reference frame also depend on higher order moments.

Energy solutions of KPZ are unique

The Kardar–Parisi–Zhang (KPZ) equation is conjectured to universally describe the fluctuations of weakly asymmetric interface growth. Here we provide the first intrinsic well-posedness result for the stationary KPZ equation on the real line by showing that itsenergy solutions, as introduced by Gonçalves and Jara in 2010 and refined by Gubinelli and Jara, are unique. This is the first time that a singular stochastic PDE can be tackled using probabilistic methods, and the combination of the convergence results of the first work and many follow-up papers with our uniqueness proof establishes the weak KPZ universality conjecture for a wide class of models. Our proof builds on an observation of Funaki and Quastel from 2015, and a remarkable consequence is that the energy solution to the KPZ equation is not equal to the Cole–Hopf solution, but it involves an additional driftt/12t/12.

Paracontrolled calculus and Funaki–Quastel approximation for the KPZ equation

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 124t. 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.

KPZ equation, its renormalization and invariant measures

The Kardar-Parisi-Zhang (KPZ) equation is a stochastic partial differential equation which is ill-posed because the nonlinearity is marginally defined with respect to the roughness of the forcing noise. However, its Cole-Hopf solution, defined as the logarithm of the solution of the linear stochastic heat equation (SHE) with a multiplicative noise, is a mathematically well-defined object. In fact, Hairer [13] has recently proved that the solution of SHE can actually be derived through the Cole-Hopf transform of the solution of the KPZ equation with a suitable renormalization under periodic boundary conditions. This transformation is unfortunately not well adapted to studying the invariant measures of these Markov processes. The present paper introduces a different type of regularization for the KPZ equation on the whole line $\mathbb{R}$ or under periodic boundary conditions, which is appropriate from the viewpoint of studying the invariant measures. The Cole-Hopf transform applied to this equation leads to an SHE with a smeared noise having an extra complicated nonlinear term. Under time average and in the stationary regime, it is shown that this term can be replaced by a simple linear term, so that the limit equation is the linear SHE with an extra linear term with coefficient 1/24. The methods are essentially stochastic analytic: The Wiener-It\^o expansion and a similar method for establishing the Boltzmann-Gibbs principle are used. As a result, it is shown that the distribution of a two-sided geometric Brownian motion with a height shift given by Lebesgue measure is invariant under the evolution determined by the SHE on $\mathbb{R}$.

Renormalized-generalized solutions for the KPZ equation

This work introduces a new notion of solution for the KPZ equation, in particular, our approach encompasses the Cole–Hopf solution. We set in the context of the distribution theory the proposed results by Bertini and Giacomin from the mid '90s. This new approach provides a pathwise notion of solution as well as a structured approximation theory. The developments are based on regularization arguments from the theory of distributions.

Fluctuation exponents for directed polymers in the intermediate disorder regime

We derive exact fluctuation exponents for a solvable model of one-dimensional directed polymers in random environment in the intermediate scaling regime. This regime corresponds to taking the inverse temperature to zero as the size of the system goes to infinity. The exponents satisfy the KPZ scaling relation and coincide with physical predictions.In the critical case, we recover the fluctuation exponent of the Hopf-Cole solution of the KPZ equation in equilibrium and close to equilibrium.

Solving the KPZ equation

We introduce a new concept of solution to the KPZ equation which is shown to extend the classical Cole-Hopf solution. This notion provides a factorisation of the Cole-Hopf solution map into a \universal measurable map from the probability space into an explicitly described auxiliary metric space, composed with a new solution map that has very good continuity properties. The advantage of such a formulation is that it essentially provides a pathwise notion of a solution, together with a very detailed approximation theory. In particular, our construction completely bypasses the Cole-Hopf transform, thus laying the groundwork for proving that the KPZ equation describes the uctuations of systems in the KPZ universality class. As a corollary of our construction, we obtain very detailed new regularity results about the solution, as well as its derivative with respect to the initial condition. Other byproducts of the proof include an explicit approximation to the stationary solution of the KPZ equation, a well-posedness result for the Fokker-Planck equation associated to a particle diusing in a rough space-time dependent potential, and a new periodic homogenisation result for the heat equation with a space-time periodic potential. One ingredient in our construction is an example of a non-Gaussian rough path such that the area process of its natural approximations needs to be renormalised by a diverging term for the approximations to converge.