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, provided that some very weak regularity conditions are met, the range of the effective diffusivity matrix can be read off the shape of the support of the invariant measure for the periodic diffusion. In particular, this gives some easily verifiable conditions for the effective diffusivity matrix to be of full rank. We also discuss the application of our results to the homogenization of a class of elliptic and parabolic PDEs.
We study the problem of characterizing the effective (homogenized) properties of materials whose diffusive properties are modeled with random fields. Focusing on elliptic PDEs with stationary and ergodic random coefficient functions, we provide a gentle introduction to the mathematical theory of homogenization of random media. We also present numerical examples to elucidate the theoretical concepts and results.
We study the homogenization property of systems of quasi-linear PDEs of parabolic type with periodic coefficients, highly oscillating drift and highly oscillating nonlinear term. To this end, we propose a probabilistic approach based on the theory of forward-backward stochastic differential equations and introduce the new concept of ``auxiliary SDEs.''
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 $BV$.
Stemmed from the derivation of the optimal control to a stochastic linear-quadratic control problem with Markov jumps, we study one kind of backward stochastic differential equations (BSDEs) that the generator f is affected by a Markovian switching. Then, the case that the Markov chain is involved in a large state space is considered. Following the classical approach, a hierarchical approach is adopted to reduce the complexity and a singularly perturbed Markov chain is involved. We will study the asymptotic property of BSDE with the singularly perturbed Markov chain. At last, as an application…
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.
We study the asymptotic behavior of solution of semi-linear PDEs. Neither periodicity nor ergodicity will be assumed. In return, we assume that the coefficients admit a limit in Cesaro sense. In such a case, the averaged coefficients could be discontinuous. We use probabilistic approach based on weak convergence for the associated backward stochastic differential equation in the S-topology to derive the averaged PDE. However, since the averaged coefficients are discontinuous, the classical viscosity solution is not defined for the averaged PDE. We then use the notion of "$L^p-$viscosity soluti…
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.
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…
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…
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…
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…
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.
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.
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 study the homogenization problem of semi linear reflected partial differential equations (reflected PDEs for short) with nonlinear Neumann conditions. The non-linear term is a function of the solution but not of its gradient. The proof are fully probabilistic and uses weak convergence of associated reflected generalized backward differential stochastic equations (reflected GBSDEs in short).
We prove the existence and uniqueness of solution of quasilinear stochastic partial differential equations with obstacle (OSPDEs in short) in degenerate case. Using De Giorgi's iteration, we deduce the $L^p-$estimates for the time-space uniform norm of weak solutions.
We extend a recently developed method to solve semi-linear PDEs to the case of a degenerated diffusion. Being a pure Monte Carlo method it does not suffer from the so called curse of dimensionality and it can be used to solve problems that were out of reach so far. We give some results of convergence and show numerically that it is effective. Besides we numerically show that the new scheme developed can be used to solve some full non linear PDEs. At last we provide an effective algorithm to implement the scheme.
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…
In this paper, we obtain the existence and uniqueness theorem for backward stochastic differential equation driven by G-Brownian motion (G-BSDE) under degenerate case. Moreover, we propose a new probabilistic method based on the representation theorem of G-expectation and weak convergence to obtain the regularity of fully nonlinear PDE associated to G-BSDE.
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.
We study multidimensional backward stochastic differential equations (BSDEs) which cover the logarithmic nonlinearity u log u. More precisely, we establish the existence and uniqueness as well as the stability of p-integrable solutions (p > 1) to multidimensional BSDEs with a p-integrable terminal condition and a super-linear growth generator in the both variables y and z. This is done with a generator f(y, z) which can be neither locally monotone in the variable y nor locally Lipschitz in the variable z. Moreover, it is not uniformly continuous. As application, we establish the existence and …
In this paper, we study a combined system of a Fokker-Planck (FP) equation for $m^{t,μ}$ with initial $(t,μ)\in[0,T]\times L^2(\mathbb{R}^d)$, and a stochastic differential equation for $X^{t,x,μ}$ with initial $(t,x)\in[0,T]\times \mathbb{R}^d$, whose coefficients depend on the solution of FP equation. We develop a combined probabilistic and analytical method to explore the regularity of the functional $V(t,x,μ)=\mathbb{E}[Φ(X^{t,x,μ}_T,m^{t,μ}(T,\cdot))]$. Our main result states that, under a non-degenerate condition and appropriate regularity assumptions on the coefficients, the function $V…
We study a class of discrete-time random dynamical systems with compact phase space. Assuming that the deterministic counterpart of the system in question possesses a dissipation property, its linearisation is approximately controllable, and the driving noise is bounded and has a decomposable structure, we prove that the corresponding family of Markov processes has a unique stationary measure, which is exponentially mixing in the dual-Lipschitz metric. The abstract result is applicable to nonlinear dissipative PDEs perturbed by a bounded random force which affects only a few Fourier modes. We …
We consider a system of semilinear partial differential equations (PDEs) with a nonlinearity depending on both the solution and its gradient. The Neumann boundary condition depends on the solution in a nonlinear manner. The uniform ellipticity is not required to the diffusion coefficient. We show that this problem admits a viscosity solution which can be approximated by a penalization. The Lipschitz condition is required to the coefficients of the diffusion part. The nonlinear part as well as the Neumann condition are Lipschitz. Moreover, the nonlinear part is assumed monotone in the solution …
We study scaling limits of the weakly driven Zhang and the Bak-Tang-Wiesenfeld (BTW) model for self-organized criticality. We show that the weakly driven Zhang model converges to a stochastic partial differential equation (PDE) with singular-degenerate diffusion. In addition, the deterministic BTW model is shown to converge to a singular-degenerate PDE. Alternatively, the proof of the scaling limit can be understood as a convergence proof of a finite-difference discretization for singular-degenerate stochastic PDEs. This extends recent work on finite difference approximation of (deterministic)…
We propose random walks on suitably defined graphs as a framework for finescale modeling of particle motion in an obstructed environment where the particle may have interactions with the obstructions and the mean path length of the particle may not be negligible in comparison to the finescale. This motivates our study of a periodic, directed, and weighted graph embedded in ${\mathbb R}^d$ and the scaling limit of the associated continuous-time random walk $Z(t)$ on the graph's nodes, which jumps along the graph's edges with jump rates given by the edge weights. We show that the scaled process …
