Homogenization of divergence-form operators with lower order terms in random media
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Citations
Stochastic flows and stochastic differential equations
Central Limits and Homogenization in Random Media
Central limits and homogenization in random media
Einstein relation for reversible diffusions in a random environment
Numerical strategy for unbiased homogenization of random materials
References
Homogenization of Linear and Semilinear Second Order Parabolic PDEs with Periodic Coefficients: A Probabilistic Approach☆
Diffusion in turbulence
Diffusion in Random Media
Dirichlet’s Problem for an Equation with Periodic Coefficients Depending on a Small Parameter
Convection-diffusion equation with space-time ergodic random flow
Related Papers (5)
Frequently Asked Questions (10)
Q2. What is the scalar form of the Dirichlet form?
The closed bilinear form and densely defined on L2(π) corresponding to the Dirichlet form E1,ω on (Ω,G, µ) isEπ(u,v) = ∫Ω ai,jDiuDjve−2V dµ ∀u,v ∈ H1(µ).
Q3. What is the difficulty for homogenization in random media?
The difficulty for homogenization in random media lies in the resolution of the auxiliary problem, that has to be done on a suitable space.
Q4. What is the pth-power of the function f?
Ω |f |p dµ is finite, then almost every realization of f belongs the space Lploc(Rn) of functions whose pth-power is locally integrable.
Q5. What is the convergent function of t in L2(]0?
Let (P̃ ε,ωt )t>0 be the semi-group whose infinitesimal generator is A ε,ω. According to Lemma 4.1 in [36, p. 147], For any function f in L2(O), a subsequence of (P̃ ε,ωt f)ε>0 is convergent in L2(]0, T [×O) for any t >
Q6. What is the central limit theorem for the stochastic process?
Using the probabilistic representation of the solutions of parabolic and elliptic PDE, this leads to establishing a Central Limit Theorem for the stochastic process generated by a second-order partial differential operator.
Q7. What is the difficulty of solving the auxiliary problem in a general random media?
For a general random media — in contradistinction to what happens in the case of periodic media where the Poincaré inequality holds (see e.g., [24] for results under weaker hypotheses in periodic media) —, the resolution of the auxiliary problem cannot be considered in a direct way, because the needed function is not a stationary random fields.
Q8. What is the measurable function of L2(Rn)?
Rn p(t, x, y)f(y) dy, it may be proved with (5) and (6) that Pt maps L 2(Rn) into the space of continuous functions that vanish at infinity.
Q9. What is the simplest way to construct a differential operator?
The authors choose here to use a bilinear form and the resolvent, but the authors also may have construct a “differential” operator and the corresponding semi-group as in [29].eratorThe authors still assume that the first-differential order term b is equal to 0.
Q10. What is the difficulty of solving the auxiliary problem?
Solving the auxiliary problem or finding the invariant measure shows thedifficulty to study the limit behaviour of the processes associated to12∂∂xi( ai,j(x/ε, ω) ∂∂xj) + 1ε bi(x/ε, ω)∂∂xifor a general b, which so far remains an open problem.