# Homogenization of divergence-form operators with lower order terms in random media

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### Cites background from "Homogenization of divergence-form o..."

...en (Eλ,ω,H 1(ρ2 λ)) is a regular Dirichlet form. We claim that Lemma 7.1 (Eλ,ω,H 1(ρ2 λ)) is the Dirichlet form of the semigroup Tλ,ω on L2(e2 ˆλ·x−2V ω(x)dx). (Note that this fact is already used in [19] but without justiﬁcation.) Proof We ﬁrst observe that Tλ,ω is indeed a strongly continuous symmetric semigroup on L2(ρ2 λ). Let t > 0 and deﬁne the approximating bilinear forms Et,λ,ω(f,f) := 1 t ...

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...nuous paths whose semigroup is Tλ,ω. We denote its law on path space C(R +,Rd) with Pλ,ω x and Eλ,ω x for the corresponding expectation. Observe that P λ,ω x [X(0) = x] = 1 for all x. It is proved in [19] Proposition 1 that, for almost all ω’s, under Pω 0 , the canonical process satisﬁes an invariance principle with some eﬀective diﬀusivity matrix Σ. It follows from Aronson’s estimate that Σ is also t...

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...tion 2’. The main diﬃculties in extending the proofs of the previous sections to measurable coefﬁcients appear in justifying the Girsanov transform and time change arguments from Section 3. Following [19], in order to do it we shall appeal to Dirichlet form theory, as exposed in [9], and related stochastic calculus for Dirichlet processes. Observe that a direct application 33 of Dirichlet form theory ...

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### "Homogenization of divergence-form o..." refers background in this paper

...Its n infinitesimal generators D1, . . . , Dn are defined by Dif = lim h→0 Theif − f h when this limit exists in L2(µ), where (e1, . . . , en) is the canonical basis of Rn....

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