On annealed elliptic Green function estimates

We consider a random, uniformly elliptic coefficient field $a$ on the lattice $\mathbb{Z}^d$. The distribution $\langle \cdot \rangle$ of the coefficient field is assumed to be stationary. Delmotte and Deuschel showed that the gradient and second mixed derivative of the parabolic Green function $G(t,x,y)$ satisfy optimal annealed estimates which are $L^2$ resp. $L^1$ in probability, i.e. they obtained bounds on $\langle |\nabla_x G(t,x,y)|^2 \rangle^{\frac{1}{2}}$ and $\langle |\nabla_x \nabla_y G(t,x,y)| \rangle$, see T. Delmotte and J.-D. Deuschel: On estimating the derivatives of symmetric diffusions in stationary random environments, with applications to the $\nabla\phi$ interface model, Probab. Theory Relat. Fields 133 (2005), 358--390. In particular, the elliptic Green function $G(x,y)$ satisfies optimal annealed bounds. In a recent work, the authors extended these elliptic bounds to higher moments, i.e. $L^p$ in probability for all $p<\infty$, see D. Marahrens and F. Otto: {Annealed estimates on the Green function}, arXiv:1304.4408 (2013). In this note, we present a new argument that relies purely on elliptic theory to derive the elliptic estimates (see Proposition 1.2 below) for $\langle |\nabla_x G(x,y)|^2 \rangle^{\frac{1}{2}}$ and $\langle |\nabla_x \nabla_y G(x,y)| \rangle$.Comment: 15 page

Cross-over in scaling laws: A simple example from micromagnetics

Scaling laws for characteristic length scales (in time or in the model parameters) are both experimentally robust and accessible for rigorous analysis. In multiscale situations cross--overs between different scaling laws are observed. We give a simple example from micromagnetics. In soft ferromagnetic films, the geometric character of a wall separating two magnetic domains depends on the film thickness. We identify this transition from a N\'eel wall to an Asymmetric Bloch wall by rigorously establishing a cross--over in the specific wall energy

Sublinear growth of the corrector in stochastic homogenization: Optimal stochastic estimates for slowly decaying correlations

We establish sublinear growth of correctors in the context of stochastic homogenization of linear elliptic PDEs. In case of weak decorrelation and "essentially Gaussian" coefficient fields, we obtain optimal (stretched exponential) stochastic moments for the minimal radius above which the corrector is sublinear. Our estimates also capture the quantitative sublinearity of the corrector (caused by the quantitative decorrelation on larger scales) correctly. The result is based on estimates on the Malliavin derivative for certain functionals which are basically averages of the gradient of the corrector, on concentration of measure, and on a mean value property for $a$-harmonic functions

The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations

We consider uniformly elliptic coefficient fields that are randomly distributed according to a stationary ensemble of a finite range of dependence. We show that the gradient and flux $(\nabla\phi,a(\nabla \phi+e))$ of the corrector $\phi$, when spatially averaged over a scale $R\gg 1$ decay like the CLT scaling $R^{-\frac{d}{2}}$. We establish this optimal rate on the level of sub-Gaussian bounds in terms of the stochastic integrability, and also establish a suboptimal rate on the level of optimal Gaussian bounds in terms of the stochastic integrability. The proof unravels and exploits the self-averaging property of the associated semi-group, which provides a natural and convenient disintegration of scales, and culminates in a propagator estimate with strong stochastic integrability. As an application, we characterize the fluctuations of the homogenization commutator, and prove sharp bounds on the spatial growth of the corrector, a quantitative two-scale expansion, and several other estimates of interest in homogenization.Comment: 114 pages. Revised version with some new results: optimal scaling with nearly-optimal stochastic integrability on top of nearly-optimal scaling with optimal stochastic integrability, CLT for the homogenization commutator, and several estimates on growth of the extended corrector, semi-group estimates, and systematic error

Quasilinear SPDEs via rough paths

We are interested in (uniformly) parabolic PDEs with a nonlinear dependance of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable: \begin{equation*} \partial_2u -P( a(u)\partial_1^2u - \sigma(u)f ) =0 \end{equation*} where $P$ is the projection on mean-zero functions, and $f$ is a distribution and only controlled in the low regularity norm of $C^{\alpha-2}$ for $\alpha > \frac{2}{3}$ on the parabolic H\"older scale. The example we have in mind is a random forcing $f$ and our assumptions allow, for example, for an $f$ which is white in the time variable $x_2$ and only mildly coloured in the space variable $x_1$; any spatial covariance operator $(1 + |\partial_1|)^{-\lambda_1 }$ with $\lambda_1 > \frac13$ is admissible. On the deterministic side we obtain a $C^\alpha$-estimate for $u$, assuming that we control products of the form $v\partial_1^2v$ and $vf$ with $v$ solving the constant-coefficient equation $\partial_2 v-a_0\partial_1^2v=f$. As a consequence, we obtain existence, uniqueness and stability with respect to $(f, vf, v \partial_1^2v)$ of small space-time periodic solutions for small data. We then demonstrate how the required products can be bounded in the case of a random forcing $f$ using stochastic arguments. For this we extend the treatment of the singular product $\sigma(u)f$ via a space-time version of Gubinelli's notion of controlled rough paths to the product $a(u)\partial_1^2u$, which has the same degree of singularity but is more nonlinear since the solution $u$ appears in both factors. The PDE ingredient mimics the (kernel-free) Krylov-Safanov approach to ordinary Schauder theory.Comment: 65 page