A Liouville theorem for stationary and ergodic ensembles of parabolic systems

Bella, Peter; Chiarini, Alberto; Fehrman, Benjamin

doi:10.1007/s00440-018-0843-z

Cited by 10 publications

(15 citation statements)

References 25 publications

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“…Let us interpret this finding in yet another way: If we are interested in predicting ∇v in the neighborhood {|x − y| < 1} of some point y with |y| ≫ 1, in view of (10), we need to know φ * i and ψ * ij in the neighborhood of y, and we need to know v h . In order to get v h , in view of (11) and (12), we need to know φ i and ψ ij in the neighborhood {|x| < 1} of the origin. Hence the local knowledge of the first and second-order correctors near the two distant points y and 0 is enough to understand how the random medium transmits the information of the charge distribution ∇ · g in {|x| < 1} to the field −∇v near y.…”

Section: The Main Resultsmentioning

confidence: 99%

Effective multipoles in random media

Bella

Giunti

Otto

2020

Communications in Partial Differential Equations

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In a homogeneous medium, the far-field generated by a localized source can be expanded in terms of multipoles; the coefficients are determined by the moments of the localized charge distribution. We show that this structure survives to some extent for a random medium in the sense of quantitative stochastic homogenization: In three space dimensions, the effective dipole and quadrupole -but not the octupole -can be inferred without knowing the realization of the random medium far away from the (overall neutral) source and the point of interest.Mathematically, this is achieved by using the two-scale expansion to higher order to construct isomorphisms between the hetero-and homogeneous versions of spaces of harmonic functions that grow at a certain rate, or decay at a certain rate away from the singularity (near the origin); these isomorphisms crucially respect the natural pairing between growing and decaying harmonic functions given by the second Green's formula. This not only yields effective multipoles (the quotient of the spaces of decaying functions) but also intrinsic moments (taken with respect to the elements of the spaces of growing functions). The construction of these rigid isomorphisms relies on a good (and dimension-dependent) control on the higher-order correctors and their flux potentials.

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Section: The Main Resultsmentioning

confidence: 99%

Effective multipoles in random media

Bella

Giunti

Otto

2020

Communications in Partial Differential Equations

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“…The first ingredient is the classical L 2 theory of parabolic systems in form of localized energy type estimates, see Lemmas 1 and 2. The second ingredient is the large-scale regularity theory for parabolic systems developed in [11] that we recall and extend in Appendix B. It provides, in particular, a large-scale C 0,1 estimate: for all x ∈ R d , there exists a stationary random variable r * (x) ≥ 1 such that for all t ∈ R and weak solution v of, for R ≥ r * (x), This properties can be used provided r * has good moment bounds, which have already been established in [15] in our context.…”

Section: Strategy Of the Proofmentioning

confidence: 99%

“…In this section we recall the regularity theory for random parabolic operator of the form ∂ τ − ∇ • a∇ developed in the papers [11,7] and draw some useful consequences. Here, we assume that a does not depend on time.…”

Section: A Probabilistic Toolsmentioning

confidence: 99%

Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields

Clozeau¹

2021

Preprint

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We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the probability space. In this contribution, we use a parabolic approach and optimally quantify the time decay of the semigroup. For the theoretical point of view, we prove an optimal decay estimate of the gradient and flux of the corrector when spatially averaged over a scale R ≥ 1. For the numerical point of view, our results provide convenient tools for the analysis of various numerical methods.

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“…Another recent result under moment conditions is a Liouville theorem for the elliptic equation associated to (1.1) in [11], cf. also [10] for a related result on the parabolic equation associated to a time-dynamic, uniformly elliptic version of (1.1). Local boundedness and a Harnack inequality for solutions to the elliptic equation were recently proven in [13] under moment conditions.…”

Section: Introductionmentioning

confidence: 99%

Off-Diagonal Heat Kernel Estimates for Symmetric Diffusions in a Degenerate Ergodic Environment

Taylor¹

2021

Preprint

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We study a symmetric diffusion process on R d , d ≥ 2, in divergence form in a stationary and ergodic random environment. The coefficients are assumed to be degenerate and unbounded but satisfy a moment condition. We derive upper off-diagonal estimates on the heat kernel of this process for general speed measure. Lower off-diagonal estimates are also shown for a natural choice of speed measure, under an additional mixing assumption on the environment. Using these estimates, a scaling limit for the Green's function is proven.

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A Liouville theorem for stationary and ergodic ensembles of parabolic systems

Cited by 10 publications

References 25 publications

Effective multipoles in random media

Effective multipoles in random media

Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields

Off-Diagonal Heat Kernel Estimates for Symmetric Diffusions in a Degenerate Ergodic Environment

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