Rémi Rhodes
We study a diffusion process with random space-time dependent coefficients. Moreover the diffusion matrix is allowed to degenerate. An invariance principle is proved provided that the diffusion coefficient is controlled by a time independent one.
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Homogenization theory of random walks in degenerate random environment
本文综述了退化随机环境中随机游走的均匀化理论，重点介绍在平稳遍历且可能无界的电导条件下建立的泛函中心极限定理、局部极限定理及热核估计。
This paper deals with homogenization of second order divergence form parabolic operators with locally stationary coefficients. Roughly speaking, locally stationary coefficients have two evolution scales: both an almost constant microscopic one and a smoothly varying macroscopic one. The homogenization procedure aims to give a macroscopic approximation that takes into account the microscopic heterogeneities. This paper follows "Diffusion in a locally stationary random environment" (published in Probability Theory and Related Fields) and improves this latter work by considering possibly degenera…
In this paper a semilinear elliptic PDE with rapidly oscillating coefficients is homogenized. The novetly of our result lies in the fact that we allow the second order part of the differential operator to be degenerate in some portion of $\R^d$.\\ Our fully probabilistic method is based on the connection between PDEs and BSDEs with random terminal time and the weak convergence of a class of diffusion processes.
We study the long time behavior (homogenization) of a diffusion in random medium with time and space dependent coefficients. The diffusion coefficient may degenerate. In Stochastic Process. Appl. (2007) (to appear), an invariance principle is proved for the critical rescaling of the diffusion. Here, we generalize this approach to diffusions whose space-time scaling differs from the critical one.
We study discrete linear divergence-form operators with random coefficients, also known as the random conductance model. We assume that the conductances are bounded, independent and stationary; the law of a conductance may depend on the orientation of the associated edge. We give a simple necessary and sufficient condition for the relaxation of the environment seen by the particle to be diffusive, in the sense of every polynomial moment. As a consequence, we derive polynomial moment estimates on the corrector.
We study the limit behaviour of a sequence of non-convex, vectorial, random integral functionals, defined on $W^{1,1}$, whose integrands satisfy degenerate linear growth conditions. These involve suitable random, scale-dependent weight-functions. Under minimal assumptions on the integrand and on the weight-functions, we show that the sequence of functionals homogenizes to a non-degenerate functional defined on $BV$.
We consider the homogenization of Hamilton-Jacobi equations and degenerate Bellman equations in stationary, ergodic, unbounded environments. We prove that, as the microscopic scale tends to zero, the equation averages to a deterministic Hamilton-Jacobi equation and study some properties of the effective Hamiltonian. We discover a connection between the effective Hamiltonian and an eikonal-type equation in exterior domains. In particular, we obtain a new formula for the effective Hamiltonian. To prove the results we introduce a new strategy to obtain almost sure homogenization, completing a pro…
This paper studies homogenization of symmetric non-local Dirichlet forms with $α$-stable-like jumping kernels in one-parameter stationary ergodic environment. Under suitable conditions, we establish homogenization results and identify the limiting effective Dirichlet forms explicitly. The coefficients of the jumping kernels of Dirichlet forms and symmetrizing measures are allowed to be degenerate and unbounded; and the coefficients in the effective Dirichlet forms can be degenerate.
It is well-known under the name of `periodic homogenization' that, under a centering condition of the drift, a periodic diffusion process on R^d converges, under diffusive rescaling, to a d-dimensional Brownian motion. Existing proofs of this result all rely on uniform ellipticity or hypoellipticity assumptions on the diffusion. In this paper, we considerably weaken these assumptions in order to allow for the diffusion coefficient to even vanish on an open set. As a consequence, it is no longer the case that the effective diffusivity matrix is necessarily non-degenerate. It turns out that, pro…
We study homogenization properties of the discrete Laplace operator with random conductances on a large domain in $\mathbb{Z}^d$. More precisely, we prove almost-sure homogenization of the discrete Poisson equation and of the top of the Dirichlet spectrum. We assume that the conductances are stationary, ergodic and nearest-neighbor conductances are positive. In contrast to earlier results, we do not require uniform ellipticity but certain integrability conditions on the lower and upper tails of the conductances. We further allow jumps of arbitrary length. Without the long-range connections, th…
We study the continuum limit of discrete, nonconvex energy functionals defined on crystal lattices in dimensions $d\geq 2$. Since we are interested in energy functionals with random (stationary and ergodic) pair interactions, our problem corresponds to a stochastic homogenization problem. In the non-degenerate case, when the interactions satisfy a uniform $p$-growth condition, the homogenization problem is well-understood. In this paper, we are interested in a degenerate situation, when the interactions neither satisfy a uniform growth condition from above nor from below. We consider interacti…
This paper deals with the homogenization problem of one-dimensional pseudo-elliptic equations with a rapidly varying random potential. The main purpose is to characterize the homogenization error (random fluctuations), i.e., the difference between the random solution and the homogenized solution, which strongly depends on the autocovariance property of the underlying random potential. It is well known that when the random potential has short-range dependence, the rescaled homogenization error converges in distribution to a stochastic integral with respect to standard Brownian motion. Here, we …
We study the random conductance model on the lattice $\mathbb{Z}^d$, i.e. we consider a linear, finite-difference, divergence-form operator with random coefficients and the associated random walk under random conductances. We allow the conductances to be unbounded and degenerate elliptic, but they need to satisfy a strong moment condition and a quantified ergodicity assumption in form of a spectral gap estimate. As a main result we obtain in dimension $d\geq 3$ quantitative central limit theorems for the random walk in form of a Berry-Esseen estimate with speed $t^{-\frac 1 5+\varepsilon}$ for…
In this article, we obtain a functional CLT for a class of degenerate diffusion processes with periodic coefficients, thus generalizing the already classical results in the context of uniformly elliptic diffusions. As an application, we also discuss periodic homogenization of a class of linear degenerate elliptic and parabolic PDEs.
We prove explicit estimates for the error in random homogenization of degenerate, second-order Hamilton-Jacobi equations, assuming the coefficients satisfy a finite range of dependence. In particular, we obtain an algebraic rate of convergence with overwhelming probability under certain structural conditions on the Hamiltonian.
Quantitative homogenization on time-dependent random conductance models with stable-like jumps
本文研究了时间依赖随机导电模型中具有稳定型跳跃的定量同质化问题，通过L^2估计和多尺度Poincaré不等式分析了随机系数的同质化性质。
We study random homogenization of second-order, degenerate and quasilinear Hamilton-Jacobi equations which are positively homogeneous in the gradient. Included are the equations of forced mean curvature motion and others describing geometric motions of level sets as well as a large class of viscous, non-convex Hamilton-Jacobi equations. The main results include the first proof of qualitative stochastic homogenization for such equations. We also present quantitative error estimates which give an algebraic rate of homogenization.
This paper concerns the homogenization problem of a parabolic equation with large, time-dependent, random potentials in high dimensions $d\geq 3$. Depending on the competition between temporal and spatial mixing of the randomness, the homogenization procedure turns to be different. We characterize the difference by proving the corresponding weak convergence of Brownian motion in random scenery. When the potential depends on the spatial variable macroscopically, we prove a convergence to SPDE.
Perturbations of Parabolic Equations and Diffusion Processes with Degeneration: Boundary Problems, Metastability, and Homogenization
本文研究了在边界上停止或反射的退化扩散过程及其带小参数抛物型方程，揭示了扰动趋于零时的亚稳态效应，并分析了 Dirichlet 与 Neumann 边界条件下的初边值问题及周期均匀化。
We aim to generalize the homogenisation theorem in \cite{Gehringer-Li-tagged} for a passive tracer interacting with a fractional Gaußian noise to also cover fractional non-Gaußian noises. To do so we analyse limit theorems for normalized functionals of Hermite-Volterra processes, extending the result in \cite{Diu-Tran} to power series with fast decaying coefficients. We obtain either convergence to a Wiener process, in the short-range dependent case, or to a Hermite process, in the long-range dependent case. Furthermore, we prove convergence in the multivariate case with both, short and long-r…
We study the qualitative homogenization of second order viscous Hamilton-Jacobi equations in space-time stationary ergodic random environments. Assuming that the Hamiltonian is convex and superquadratic in the momentum variable (gradient) we establish a homogenization result and characterize the effective Hamiltonian for arbitrary (possibly degenerate) elliptic diffusion matrices. The result extends previous work that required uniform ellipticity and space-time homogeneity for the diffusion.
Quantitative homogenization for log-normal coefficients
针对既非有界也非一致椭圆的对数正态系数，本文借鉴非线性情形方法并结合退化系数的大尺度正则性，建立了关于振荡与涨落的最优定量均匀化结果。
We analyze the solutions of the Schrödinger equation with the low frequency initial data and a time-dependent weakly random potential. We prove a homogenization result for the low frequency component of the wave field. We also show that the dynamics generates a non-trivial energy in the high frequencies, which do not homogenize -- the high frequency component of the wave field remains random and the evolution of its energy is described by a kinetic equation. The transition from the homogenization of the low frequencies to the random limit of the high frequencies is illustrated by understanding…
We consider a system of differential equations in a fast long range dependent random environment and prove a homogenization theorem involving multiple scaling constants. The effective dynamics solves a rough differential equation, which is `equivalent' to a stochastic equation driven by mixed Itô integrals and Young integrals with respect to Wiener processes and Hermite processes. Lacking other tools we use the rough path theory for proving the convergence, our main technical endeavour is on obtaining an enhanced scaling limit theorem for path integrals (Functional CLT and non-CLT's) in a stro…
Let ${\bf{w}}(t,x):=(u,v)(t,x),\ t>0,\ x\in \mathbb{R}^{n},$ be the $\mathbb{R}^2$-valued spatial-temporal random field ${\bf{w}}=(u, v)$ arising from a certain two-equation system of fractional kinetic equations of reaction-diffusion type, with given random initial data $u(0,x)$ and $v(0,x).$ The space-fractional derivative is characterized by the composition of the inverses of the Riesz potential and the Bessel potential. We discuss two scaling limits, the macro and the micro, for the homogenization of ${\bf{w}}(t,x)$, and prove that the rescaled limit is a singular field of multiple Itô-Wie…
Homogenization theory of random walks among deterministic conductances
本文研究了确定性导电环境中随机游走的均质化理论，确定了满足不变原理的导电配置集，并为其他运动的去随机化均质化提供了框架。
We consider an energy conserving linear dynamics that we perturb by a Glauber dynamics with random site dependent intensity. We prove hydrodynamic limits for this non-reversible system in random media. The diffusion coefficient turns out to depend on the random field only by its statistics. The diffusion coefficient defined through the Green-Kubo formula is also studied and its convergence to some homogenized diffusion coefficient is proved.
We prove that diffusion equations with a space-time stationary and ergodic, divergence-free drift homogenize in law to a deterministic stochastic partial differential equation with Stratonovich transport noise. In the absence of spatial ergodicity, the drift is only partially absorbed into the skew-symmetric part of the flux through the use of an appropriately defined stream matrix. This leaves a time-dependent, spatially-homogenous transport which, for mildly decorrelating fields, converges to a Brownian noise with deterministic covariance in the homogenization limit. The results apply to uni…
It is well-known that the central limit theorem holds for partial sums of a stationary sequence $(X_i)$ of $m$-dependent random variables with finite variance; however, the limit may be degenerate with variance 0 even if $\mathrm{Var}(X_i)\neq0$. We show that this happens only in the case when $X_i-\mathbb E X_i=Y_i-Y_{i-1}$ for an $(m-1)$-dependent stationary sequence $(Y_i)$ with finite variance (a result implicit in earlier results), and give a version for block factors. This yields a simple criterion that is a sufficient condition for the limit not to degenerate. Two applications to subtre…
In terms of a nice reference probability measure, integrability conditions on the path-dependent drift are presented for (infinite-dimensional) degenerate PDEs to have regular positive solutions. To this end, the corresponding stochastic (partial) differential equations are proved to possess the weak existence and uniqueness of solutions, as well as the existence, uniqueness and entropy estimates of invariant probability measures. When the reference measure satisfies the log-Sobolev inequality, Sobolev estimates are derived for the density of invariant probability measures. Some results are ne…
Degenerate Diffusions on Continuum Percolation and Hamilton-Jacobi-Bellman Equations
研究在连续渗流簇上的退化受控扩散和Hamilton-Jacobi-Bellman方程，证明扩散永不触及边界并存在唯一全局强解，建立随机控制表示公式，验证布尔模型等具体模型满足假设。
We study homogenization for a class of generalized Langevin equations (GLEs) with state-dependent coefficients and exhibiting multiple time scales. In addition to the small mass limit, we focus on homogenization limits, which involve taking to zero the inertial time scale and, possibly, some of the memory time scales and noise correlation time scales. The latter are meaningful limits for a class of GLEs modeling anomalous diffusion. We find that, in general, the limiting stochastic differential equations (SDEs) for the slow degrees of freedom contain non-trivial drift correction terms and are …
To characterize nonlinear Dirichlet problems in an open domain, we investigate killed distribution dependent SDEs. By constructing the coupling by projection and using the Zvonkin/Girsanov transforms, the well-posedness is proved for three different situations: 1) monotone case with distribution dependent noise (possibly degenerate), 2) singular case with non-degenerate distribution dependent noise, and 3) singular case with non-degenerate distribution independent noise. In the first two cases the domain is $C^2$ smooth such that the Lipschitz continuity in initial distributions is also derive…
By comparing the original equations with the corresponding stationary ones, the moderate deviation principle (MDP) is established for unbounded additive functionals of several different models of distribution dependent SDEs, with non-degenerate and degenerate noises.
We study a continuous-time random walk, $X$, on $\mathbb{Z}^d$ in an environment of dynamic random conductances taking values in $(0, \infty)$. We assume that the law of the conductances is ergodic with respect to space-time shifts. We prove a quenched invariance principle for the Markov process $X$ under some moment conditions on the environment. The key result on the sublinearity of the corrector is obtained by Moser's iteration scheme.
We consider continuous-time random walks on a random locally finite subset of $\mathbb{R}^d$ with random symmetric jump probability rates. The jump range can be unbounded. We assume some second--moment conditions and that the above randomness is left invariant by the action of the group $\mathbb{G}=\mathbb{R}^d$ or $\mathbb{G}=\mathbb{Z}^d$. We then add a site-exclusion interaction, thus making the particle system a simple exclusion process. We show that, for almost all environments, under diffusive space-time rescaling the system exhibits a hydrodynamic limit in path space. The hydrodynamic e…
We study a class of systems whose dynamics are described by generalized Langevin equations with state-dependent coefficients. We find that in the limit, in which all the characteristic time scales vanish at the same rate, the position variable of the system converges to a homogenized process, described by an equation containing additional drift terms induced by the noise. The convergence results are obtained using the main result in \cite{hottovy2015smoluchowski}, whose version is proven here under a weaker spectral assumption on the damping matrix. We apply our results to study thermophoresis…
We consider a stationary and ergodic random field {ω(b)} parameterized by the family of bonds b in Z^d, d>1. The random variable ω(b) is thought of as the conductance of bond b and it ranges in a finite interval [0,c_0]. Assuming that the set of bonds with positive conductance has a unique infinite cluster C, we prove homogenization results for the random walk among random conductances on C. As a byproduct, applying the general criterion of \cite{F} leading to the hydrodynamic limit of exclusion processes with bond-dependent transition rates, for almost all realizations of the environment we p…
We provide an example of an SDE with degenerate additive noise where synchronization depends on the strength of noise and the number of directions in which the noise acts. Here, synchronization means that the weak random attractor consists of a single random point. Indicated by a change of sign of the top Lyapunov exponent, we prove synchronization respectively no (weak) synchronization.
We prove a comparison result for viscosity solutions of (possibly degenerate) parabolic fully nonlinear path-dependent PDEs. In contrast with the previous result in Ekren, Touzi & Zhang, our conditions are easier to check and allow for the degenerate case, thus including first order path-dependent PDEs. Our argument follows the regularization method as introduced by Jensen, Lions & Souganidis in the corresponding finite-dimensional PDE setting. The present argument significantly simplifies the comparison proof in Ekren, Touzi & Zhang, but requires an $L^p-$type of continuity (with respect to t…
Weak Existence for Degenerate Distribution Dependent SDEs with multiplicative Noise -- a pathwise regularization approach
本研究将现有弱存在性技术拓展至分布依赖框架，结合路径平均与局部时分解方法，证明了带退化乘性噪声和奇异系数的分布依赖随机微分方程弱解的存在性，揭示了不规则噪声对随机系统退化相关分析难题的正则化作用。
We establish a quenched local central limit theorem for the dynamic random conductance model on $\mathbb{Z}^d$ only assuming ergodicity with respect to space-time shifts and a moment condition. As a key analytic ingredient we show Hölder continuity estimates for solutions to the heat equation for discrete finite difference operators in divergence form with time-dependent degenerate weights. The proof is based on De Giorgi's iteration technique. In addition, we also derive a quenched local central limit theorem for the static random conductance model on a class of random graphs with degenerate …
Tiny fluctuations of the averaging process around its degenerate steady state
研究了平均过程在退化稳态附近的小波动，通过非标准CLT缩放因子和扩散性重新缩放，揭示了高斯波动的产生机制，并利用皮卡不等式和梯度矩估计得到方差界。
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