arXiv subjects

Tadahisa Funaki

Publications and source records attributed to Tadahisa Funaki.

At least 19 recordsLinked to original sources

arXiv2025

$L^p$-Boltzmann-Gibbs principle via Littlewood-Paley-Stein inequality

In this paper, we establish the Boltzmann-Gibbs principle in the $L^p$ sense by applying the Littlewood-Paley-Stein inequality. Our model is an asymmetric Ginzburg-Landau interface model on a one-dimensional periodic lattice. Assuming convexity of the potential,we derive detailed error estimates, particularly their dependence on the size of the system and the size of the region on which the sample average is taken. Notably, the estimates are uniform in the strength of the asymmetry.

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arXiv2025

Coupled KPZ equations and their decoupleability

We discuss characterizations of the decoupleability, partial and full, of trilinear or completely symmetric real $n\times n\times n$ tensors, which inform on the structure of certain coupled KPZ equations. Informally, when the tensor is partially decoupleable, one of the components in the coupled KPZ equation splits off from the others, while when the tensor is fully decoupleable, each of the $n$ components splits off from the others. Such a characterization is recast as a problem of membership of trilinear tensors in $O(n)$ orbits of subsets of fully decoupleable and partially decoupleable tensors. When $n=2$, we show these subsets are the same, and in this case give a single criterion in terms of the entries of a tensor for membership in the orbits of these subsets. When $n\geq 3$, the subsets are different. For $n\geq 3$, we characterize full decoupleability in terms of several abstract relations, which when $n=3$ are made explicit. When $n=3$, we also explicitly characterize partial decoupleability. The methods involve notions in applied invariant theory, relating $O(n)$ invariant subsets to stabilizer subgroup actions on smaller sets. When $n=3$ make use of the explicit basis of invariants found by Olive and Auffray. When $n=2$, we also supply two other more direct arguments.

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arXiv2024

Linear fluctuation of interfaces in Glauber-Kawasaki dynamics

In this article, we find a scaling limit of the space-time mass fluctuation field of Glauber + Kawasaki particle dynamics around its hydrodynamic mean curvature interface limit. Here, the Glauber rates are scaled by $K=K_N$, the Kawasaki rates by $N^2$ and space by $1/N$. We start the process so that the interface $\Gamma_t$ formed is stationary that is, $\Gamma_t$ is `flat'. When the Glauber rates are balanced on $T^d$, $\Gamma_t=\Gamma=\{x: x_1=0\}$ is immobile and the hydrodynamic limit is given by $\rho(t,v) = \rho_+$ for $v_1\in (0,1/2)$ and $\rho(t,v)= \rho_-$ for $v_1\in (-1/2,0)$ for all $t\ge 0$, where $v=(v_1,\ldots,v_d)\in T^d$ identified with $[-1/2,1/2)^d$. Since in the formation the boundary region about the interface has width $O(1/\sqrt{K_N})$, we will scale the $v_1$ coordinate in the fluctuation field by $\sqrt{K_N}$ so that the scaling limit will capture information `near' the interface. We identify the fluctuation limit as a Gaussian field when $K_N\uparrow \infty$ and $K_N= O(\sqrt{\log(N)})$ in $d\leq 2$. In the one dimensional case, the field limit is given by ${\bf e}(v_1) B_t$ where $B_t$ is a Brownian motion and ${\bf e}$ is the normalized derivative of a decreasing `standing wave' solution $\phi$ of $\partial^2_{v_1} \phi - V'(\phi)=0$ on $R$, where $V'$ is the homogenization of the Glauber rates. In two dimensions, the limit is ${\bf e}(v_1)Z_t(v_2)$ where $Z_t$ is the solution of a one dimensional stochastic heat equation. The appearance of the function ${\bf e}(\cdot)$ in the limit field indicates that the interface fluctuation retains the shape of the transition layer $\phi$.

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arXiv2024

Stochastic PDE approach to fluctuating interfaces

We propose a new type of SPDEs, singular or with regularized noises, motivated by a study of the fluctuation of the density field in a microscopic interacting particle system. They include a large scaling parameter $N$, which is the ratio of macroscopic to microscopic size, and another scaling parameter $K=K(N)$, which controls the formation of the interface of size $K^{-1/2}$ in the density field. They are derived heuristically from the particle system, assuming the validity of the so-called ``Boltzmann-Gibbs principle", that is, a combination of the local ensemble average due to the local ergodicity and its asymptotic expansion. We study a simple situation where the interface is flat and immobile. Under making a proper stretch to the normal direction to the interface, we observe a Gaussian fluctuation of the interface. We also heuristically derive a nonlinear SPDE which describes the fluctuation of the interface.

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arXiv2024

Interface motion from Glauber-Kawasaki dynamics of non-gradient type

We consider the Glauber-Kawasaki dynamics on a $d$-dimensional periodic lattice of size $N$, that is, a stochastic time evolution of particles performing random walks with interaction subject to the exclusion rule (Kawasaki part), in general, of non-gradient type, together with the effect of the creation and annihilation of particles (Glauber part) whose rates are set to favor two levels of particle density, called sparse and dense. We then study the limit of our dynamics under the hydrodynamic space-time scaling, that is, $1/N$ in space and a diffusive scaling $N^2$ for the Kawasaki part and another scaling $K=K(N)$, which diverges slower, for the Glauber part in time. In the limit as $N\to\infty$, we show that the particles autonomously make phase separation into sparse or dense phases at the microscopic level, and an interface separating two regions is formed at the macroscopic level and evolves under an anisotropic curvature flow. In the present article, we show that the particle density at the macroscopic level is well approximated by a solution of a reaction-diffusion equation with a nonlinear diffusion term of divergence form and a large reaction term. Furthermore, by applying the results of Funaki, Gu and Wang [arXiv:2404.12234] for the convergence rate of the diffusion matrix approximated by local functions, we obtain a quantitative hydrodynamic limit as well as the upper bound for the allowed diverging speed of $K=K(N)$. The above result for the derivation of the interface motion is proved by combining our result with that in a companion paper by Funaki and Park [arXiv:2403.01732], in which we analyzed the asymptotic behavior of the solution of the reaction-diffusion equation obtained in the present article and derived an anisotropic curvature flow in the situation where the macroscopic reaction term determined from the Glauber part is bistable and balanced.

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arXiv2024

Quantitative homogenization and hydrodynamic limit of non-gradient exclusion process

For the non-gradient exclusion process, we prove the quantitative homogenization of the diffusion matrix and the conductivity by local functions. The proof relies on the renormalization approach developed by Armstrong, Kuusi, Mourrat, and Smart, while the new challenge here is the hard core constraint of particle number on every site. Therefore, a coarse-grained method is proposed to lift the configuration to a larger space without exclusion, and a gradient coupling between two systems is applied to capture the spatial cancellation. We then strengthen the convergence rate to be uniform concerning the density, and integrate it into the work by Funaki, Uchiyama, and Yau [IMA Vol. Math. Appl., 77 (1996), pp. 1-40.] to yield a quantitative hydrodynamic limit. Our new approach avoids showing the characterization of closed forms and provides stronger results. The extension is discussed for the model in the presence of disorder on the bonds.

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arXiv2024

Motion of sharp interface of Allen-Cahn equation with anisotropic nonlinear diffusion

We consider the Allen-Cahn equation with nonlinear anisotropic diffusion and derive anisotropic direction-dependent curvature flow under the sharp interface limit. The anisotropic curvature flow was already studied, but its derivation is new. We prove both generation and propagation of the interface. For the proof we construct sub- and super-solutions applying the comparison theorem. The problem discussed in this article naturally appeared in the study of the interacting particle systems, especially of non-gradient type. The Allen-Cahn equation obtained from systems of gradient type has a simpler nonlinearity in diffusion and leads to isotropic mean-curvature flow. We extend those results to anisotropic situations.

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arXiv2022

Motion by mean curvature from Glauber-Kawasaki dynamics with speed change

We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of Glauber-Kawasaki dynamics with speed change. The Kawasaki part describes the movement of particles through particle interactions. It is speeded up in a diffusive space-time scaling. The Glauber part governs the creation and annihilation of particles. The Glauber part is set to favor two levels of particle density. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, a mean-curvature interface flow emerges, with a homogenized `surface tension-mobility' parameter reflecting microscopic rates. The interface separates the two levels of particle density. Similar hydrodynamic limits have been derived in two recent papers; one where the Kawasaki part describes simple nearest neighbor interactions, and one where the Kawasaki part is replaced by a zero-range process. We extend the main results of these two papers beyond nearest-neighbor interactions. The main novelty of our proof is the derivation of a `Boltzmann-Gibbs' principle which covers a class of local particle interactions.

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arXiv2022

Constant-speed interface flow from unbalanced Glauber-Kawasaki dynamics

We derive the hydrodynamic limit of Glauber-Kawasaki dynamics. The Kawasaki part is simple and describes independent movement of the particles with hard core exclusive interactions. It is speeded up in a diffusive space-time scaling. The Glauber part describes the birth and death of particles. It is set to favor two levels of particle density with a preference for one of the two. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, the limiting particle density instantly takes either of the two favored density values. The interface which separates these two values evolves with constant speed (Huygens' principle). Similar hydrodynamic limits have been derived in four recent papers. The crucial difference with these papers is that we consider Glauber dynamics which has a preferences for one of the two favored density values. As a result, we observe limiting dynamics on a shorter time scale, and the evolution is different from the mean curvature flow obtained in the four previous papers. While several steps in our proof can be adopted from these papers, the proof for the propagation of the interface is new.

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arXiv2021

Mean curvature interface limit from Glauber+Zero-range interacting particles

We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of a class of Glauber+Zero-range particle systems. The Zero-range part moves particles while preserving particle numbers, and the Glauber part governs the creation and annihilation of particles and is set to favor two levels of particle density. When the two parts are simultaneously seen in certain different time-scales, the Zero-range part being diffusively scaled while the Glauber part is speeded up at a lesser rate, a mean-curvature interface flow emerges, with a homogenized `surface tension-mobility' parameter reflecting microscopic rates, between the two levels of particle density. We use relative entropy methods, along with a suitable `Boltzmann-Gibbs' principle, to show that the random microscopic system may be approximated by a `discretized' Allen-Cahn PDE with nonlinear diffusion. In turn, we show the behavior, especially generation and propagation of interface properties, of this `discretized' PDE.

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arXiv2021

Schauder estimate for quasilinear discrete PDEs of parabolic type

We investigate quasilinear discrete PDEs $\partial_t u = Δ^N φ(u)+ Kf(u)$ of reaction-diffusion type with nonlinear diffusion term defined on an $n$-dimensional unit torus discretized with mesh size $\tfrac1N$ for $N\in {\mathbb N}$, where $Δ^N$ is the discrete Laplacian, $φ$ is a strictly increasing $C^5$ function and $f$ is a $C^1$ function. We establish $L^\infty$ bounds and space-time Hölder estimates, both uniform in $N$, of the first and second spatial discrete derivatives of the solutions. In the equation, $K>0$ is a large constant and we show how these estimates depend on $K$. The motivation for this work stems originally from the study of hydrodynamic scaling limits of interacting particle systems. Our method is a two steps approach in terms of the Hölder estimate and Schauder estimate, which is known for continuous parabolic PDEs. We first show the discrete Hölder estimate uniform in $N$ for the solutions of the associated linear discrete PDEs with continuous coefficients, based on the Nash estimate. We next establish the discrete Schauder estimate for linear discrete PDEs with uniform Hölder coefficients. The link between discrete and continuous settings is given by the polylinear interpolations. Since this operation has a non-local nature, the method requires proper modifications. We also discuss another method based on the study of the corresponding fundamental solutions.

math.AP↗

arXiv2021

Singular limit of an Allen-Cahn equation with nonlinear diffusion

We consider an Allen-Cahn equation with nonlinear diffusion, motivated by the study of the scaling limit of certain interacting particle systems. We investigate its singular limit and show the generation and propagation of an interface in the limit. The evolution of this limit interface is governed by mean curvature flow with a novel, homogenized speed in terms of a surface tension-mobility parameter emerging from the nonlinearity in our equation.

math.AP↗

arXiv2021

Global solvability and convergence to stationary solutions in singular quasilinear stochastic PDEs

We consider singular quasilinear stochastic partial differential equations (SPDEs) studied in \cite{FHSX}, which are defined in paracontrolled sense. The main aim of the present article is to establish the global-in-time solvability for a particular class of SPDEs with origin in particle systems and, under a certain additional condition on the noise, prove the convergence of the solutions to stationary solutions as $t\to\infty$. We apply the method of energy inequality and Poincaré inequality. It is essential that the Poincaré constant can be taken uniformly in an approximating sequence of the noise. We also use the continuity of the solutions in the enhanced noise, initial values and coefficients of the equation, which we prove in this article for general SPDEs discussed in \cite{FHSX} except that in the enhanced noise. Moreover, we apply the initial layer property of improving regularity of the solutions in a short time.

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arXiv2020

Asymptotics of PDE in random environment by paracontrolled calculus

We apply the paracontrolled calculus to study the asymptotic behavior of a certain quasilinear PDE with smeared mild noise, which originally appears as the space-time scaling limit of a particle system in random environment on one dimensional discrete lattice. We establish the convergence result and show a local in time well-posedness of the limit stochastic PDE with spatial white noise. It turns out that our limit stochastic PDE does not require any renormalization. We also show a comparison theorem for the limit equation.

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arXiv2019

Derivation of coupled KPZ-Burgers equation from multi-species zero-range processes

We consider the fluctuation fields of multi-species weakly-asymmetric zero-range interacting particle systems in one dimension, where the mass density of each species is conserved. Although such fields have been studied in systems with a single species, the multi-species setting is much less understood. Among other results, we show that, when the system starts from stationary states, with a particular property, the scaling limits of the multi-species fluctuation fields, seen in a characteristic traveling frame, solve a coupled Burgers SPDE, which is a formal spatial gradient of a coupled KPZ equation.

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arXiv2019

Motion by mean curvature from Glauber-Kawasaki dynamics

We study the hydrodynamic scaling limit for the Glauber-Kawasaki dynamics. It is known that, if the Kawasaki part is speeded up in a diffusive space-time scaling, one can derive the Allen-Cahn equation which is a kind of the reaction-diffusion equation in the limit. This paper concerns the scaling that the Glauber part, which governs the creation and annihilation of particles, is also speeded up but slower than the Kawasaki part. Under such scaling, we derive directly from the particle system the motion by mean curvature for the interfaces separating sparse and dense regions of particles as a combination of the hydrodynamic and sharp interface limits.

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arXiv2019

Fast-reaction limit for Glauber-Kawasaki dynamics with two components

We consider the Kawasaki dynamics of two types of particles under a killing effect on a $d$-dimensional square lattice. Particles move with possibly different jump rates depending on their types. The killing effect acts when particles of different types meet at the same site. We show the existence of a limit under the diffusive space-time scaling and suitably growing killing rate: segregation of distinct types of particles does occur, and the evolution of the interface between the two distinct species is governed by the two-phase Stefan problem. We apply the relative entropy method and combine it with some PDE techniques.

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arXiv2017

A coupled KPZ equation, its two types of approximations and existence of global solutions

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.

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