相关论文: Solving the KPZ equation
Trefftz schemes are high-order Galerkin methods whose discrete spaces are made of elementwise exact solutions of the underlying PDE. Trefftz basis functions can be easily computed for many PDEs that are linear, homogeneous, and have…
We prove optimal regularity results for solutions to linear kinetic Fokker-Planck equations in bounded domains. Our contributions are two-fold. First, we establish the sharp $C^{1/2}$ regularity for either diffuse reflection or prescribed…
Motivated by emerging applications from imaging processing, the heat flow of a generalized $p$-harmonic map into spheres is studied for the whole spectrum, $1\leq p<\infty$, in a unified framework. The existence of global weak solutions is…
This article studies the temporal law of the KPZ fixed point. For the stationary geometry, we find the two-time law, which extends the single time law due to Baik-Rains and Ferrari-Spohn. For the droplet geometry, we find a relatively…
This paper discusses an improved smoothing phenomena for low-regularity solutions of the Korteweg-de Vries (KdV) equation in the periodic settings by means of normal form transformation. As a result, the solution map from a ball on…
The central aim of this work is to understand rough differential equations on homogeneous spaces. We focus on the formal approach, by giving an explicit expansion of the solution at each point of the real line in terms of decorated planar…
For every strong coarse homology theory we construct a coarse assembly map as a natural transformation between coarse homology theories. We provide various conditions implying that this assembly map is an equivalence. These results…
The celebrated Kardar-Parisi-Zhang (KPZ) equation describes the kinetic roughening of stochastically growing interfaces. In one dimension, the KPZ equation is exactly solvable and its statistical properties are known to an exquisite degree.…
This paper is devoted to a fundamental solution of a nonlinear kinetic equation involving a porous medium or fast diffusion operator acting on velocities. Such a nonlinearity has interesting scaling properties, which result in a…
We develop a potential theory for the Wess--Zumino--Witten (WZW) equation in the space of K\"ahler potentials which is parallel to the potential theory for the Hermitian--Yang--Mills equation. A concept called $\omega$-harmonicity on graphs…
The periodic KPZ fixed point is the conjectural universal limit of the KPZ universality class models on a ring when both the period and time critically tend to infinity. For the case of the periodic narrow wedge initial condition, we…
The Fokker-Planck equation can be reformulated as a continuity equation, which naturally suggests using the associated velocity field in particle flow methods. While the resulting probability flow ODE offers appealing properties - such as…
An idea of Hopf's for applying complex analysis to the study of constant mean curvature spheres is generalized to cover a wider class of spheres, namely, those satisfying a Weingarten relation of a certain type, namely H = f(H^2-K) for some…
Spurred by theoretical predictions from Spohn and coworkers [Phys. Rev. E {\bf 69}, 035102(R) (2004)], we rederived and extended their result heuristically as well as investigated the scaling properties of the associated Langevin equation…
Anomalous diffusion and power-law distributions are observed in various complex systems. To provide a consistent dynamical foundation for these phenomena, we present a geometric derivation of the nonlinear Fokker-Planck equation by…
We investigate a family of generalized Fokker-Planck equations that contains Richardson and porous media equations as members. Considering a confining drift term that is related to an effective potential, we show that each equation of this…
We consider a plane wave, a radiation solution, and the sum of these solutions (total solution) for the Helmholtz equation in an exterior region in $\mathbb R^2$. We consider a straight line in this region, such that the direction of…
We discuss a recent application of the Modified Homotopy Perturbation Method to the Fokker--Planck equation and show that the selected examples do not have any connection with actual physical problems.
Opposing to a (common) belief against the existence of a thermodynamic-like potential for the KPZ equation, here we present a derivation for such a functional. With its knowledge we prove some global shift invariance properties previously…
We present a complete quantization of an approximately homogeneous and isotropic universe with small scalar perturbations. We consider the case in which the matter content is a minimally coupled scalar field and the spatial sections are…