Variational Convergence of Finite Networks

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64 KASUE if limn!1 supXn ju fn unj ¼0. Let F : CðYÞ !½0; þ1Š and F n : CðXnÞ !½0; þ1Š be lower semi-continuous functionals on CðYÞ and CðXnÞ respectively. We say that F n -converges to F if the following conditions are satisfied: (i) if a sequence of functions un 2 CðXnÞ uniformly converges to a function u 2 CðYÞ, then we have F ðuÞ lim inf n!1 F nðunÞ ð þ1Þ; (ii) for any u 2 CðYÞ, there exists a sequence of functions un 2 CðXnÞ such that un uniformly converges to u and lim sup F nðunÞ n!1 F ðuÞ ð þ1Þ: The following is a basic fact on this variational convergence. Theorem 3.3. Let Y and fXng be respectively a compact, separable, Hausdorff space and a sequence of such spaces. Given a sequence of maps fn : Xn ! Y and a sequence of lower semi-continuous functionals F n : CðXnÞ !½0; þ1Š, there exsists a subsequence, F m, and a lower semi-continuous functional F : CðYÞ !½0; þ1Š such that F m converges to F as m !1. - Proof. Using the idea of De Giorgi’s -convergence (cf. e.g. [5],), we introduce a functional on CðYÞ as follows: Let B ¼fOig be a countable basis of CðYÞ such that Oi is totally bounded. Given Oi and a positive integer k, let Oi;k;n ¼fv 2 CðXnÞ jsup ju Xn fn vj < 1=k for some u 2 Oig; and Ei;k;n ¼ inffF nðvÞ jv 2 Oi;k;ng ð þ1Þ: Then passing to a subsequence, fXmg, we may assume that for any Oi and every k, Ei;k;m tends to an extended number Ei;k 2½0; þ1Š as m !1, and thus we are able to obtain a lower semi-continuous functional F : CðYÞ !½0; þ1Š defined by F ðuÞ ¼supfEi;k j u 2 Oi; k > 0g; u 2 CðYÞ; to which F m -converges as m !1. Let F n : CðXnÞ !½0; þ1Š and F : CðYÞ !½0; þ1Š be as in Theorem 3.3. If F n is induced from a quadratic form En on a subspace D½EŠ of CðXnÞ, that is, F ðuÞ ¼Eðu; uÞ for u 2 D½EŠ and F ðuÞ ¼þ1for u 2 CðXnÞnD½EŠ, then the - limit F : CðYÞ !½0; 1Þ is also induced from a quadratic form E on a subspace D½EŠ of CðYÞ. For the functional induced from a quadratic frm E : D½EŠ D½EŠ !R, we do not distinguish between the functional and the form E. If all F n satisfy the property that F nðuÞ F ðuÞ, u 2 CðXnÞ, then so does the -limit F , where for a continuous function u 2 CðXnÞ, we set u ¼ maxf0; minfu; 1gg. 3.3 Now we state some results. Let fðXn; RnÞg be a sequence of compact metric spaces of resistance forms En. We assume that the following conditions are satisfied: (i) There exist a compact, separable Hausdorff space Y, a sequence of maps fn : Xn ! Y and also a sequence of maps hn : Y ! Xn such that fn hn uniformly converges to the identity map of Y as n !1. (ii) The functional En -converges (via fn) to a functional E : CðYÞ !½0; þ1Š as n !1. (iii) For any sequence of functions un 2 D½EnŠ such that supn maxXn junj < þ1 and supn Enðun; unÞ < þ1, there exists a subsequence fumg which uniformly converges to a function u 2 CðYÞ as m !1. Under these conditions, our main result is stated in Theorem 3.4. (iv) Let The following assertions hold: juðxÞ uðyÞj2 REðx; yÞ ¼sup j u 2 D½EŠ; Eðu; uÞ 6¼ 0 ; Eðu; uÞ x; y 2 Y: Then RE : Y Y !½0; þ1Š induces a continuous pseudo-distance on Y (admitting þ1 in its values), and one has 0 REðx; yÞ ¼ lim n!1 RnðhnðxÞ; hnðyÞÞ þ1; x; y 2 Y: (v) On Y, an equivalence relation b is introduced as follows: x b y if and only if REðx; yÞ < þ1. Then Y is decomposed into a finite number of the equivalence classes Y ð ¼ 1; ...; pÞ; each class Y is open and closed in Y. Moreover for each and large n, the inverse image of Y by fn, Xn; ¼ f 1 n ðY Þ, is open and closed in Xn and one has lim n!1 supfjRnðx; yÞ REð fnðxÞ; fnðyÞÞj j x; y 2 Xn; g¼0; lim REð fnðXn; Þ; yÞ ¼0: n!1 sup y2Y
Variational Convergence of Finite Networks 65 (vi) Let be the characteristic function of the subspace Y ð1 pÞ. Then 2 D½EŠ and Eð ; uÞ ¼0 for all u 2 D½EŠ, and if Eðu; uÞ ¼0, then u is a linear combination of the characteristic functions ð1 pÞ. Moreover let D½E Š¼fu 2 D½EŠ jsupp u Y g and E ðu; vÞ ¼Eðu; vÞ ðu; v 2 D½E ŠÞ. Then one has D½EŠ ¼ Pp ¼1 D½E Š and Eðu; vÞ ¼ Pp ¼1 E ð u; vÞ (vii) Another equivalence relation 0 is defined on Y by x 0 y if and only if REðx; yÞ ¼0. Let Y ¼ Y= 0 and Y ¼ Y = 0 ð ¼ 1; ...; pÞ be respectively the quotient spaces of Y and Y . Then for each , RE provides Y a distance R which induces the same topology as the original one, and D½E Š is included in the pull-back of CðY Þ by the canonical projection of Y onto Y . Thus the form E can be assumed to be defined on CðY Þ; ðE ; D½E ŠÞ becomes a resistance form on Y and R is the associated resistance metric, that is, juðx Þ uðy Þj2 R ðx ; y Þ¼sup j u 2 D½E Š; E ðu; uÞ 6¼ 0 ; E ðu; uÞ x ; y 2 Y : Moreover a sequence of the compact metric spaces ðXn; ; RnÞ converges to ðY ; R Þ as n !1 in the Gromov- Hausdorff sense via the approximating maps fn : Xn; ! Y , and the form EXn; on CðXn; n !1. Þ -converges to E as We remark that the compactness condition (iii) as above is indispensable in Theorem 3.4, and some results follows from the theorem. Corollary 3.5. Let fKng be a sequence of closed subspaces of compact metric graphs Gn and suppose that the metric space Kn with the induced Riemannian distance dR Gn converges to a compact metric space ðX; dXÞ in the Gromov- Hausdorff sense via approximating maps fn : Kn ! X. Then the resistance metric RGn on Kn converges to a continuous pseodo-distance R on X with respect to the Gromov-Hausdorff distance via the same approximating maps if and only if the resistance form EKn has -converges to a lower semi-continuous functional E on CðXÞ as n !1. In these cases, one 0 Rðx; yÞ dXðx; yÞ; x; y 2 X; and R is given by Rðx; yÞ ¼sup juðxÞ uðyÞj2 Eðu; uÞ j u 2 D½EŠ; Eðu; uÞ 6¼ 0 ; x; y 2 X: Corollary 3.6. Let fðXn; RnÞg be a sequence of compact metric spaces of resistance forms En which converges to a compact metric space ðY; RYÞ in the Gromov-Hausdorff sense via approximating maps fn : Xn ! Y. Then the resistance form E -converges to a resistance form E on CðXÞ which is associated with the limit distance RY. Kn In these corollaries, the definition of a resistance metric allows us to apply Ascoli-Arzelà’s theorem to the sequences and see that the compactness condition (iii) holds true. In the case where a sequence of networks ðV; En; rnÞ with the same set of verticies V is considered, letting fn as in Theorem 3.4 be the identity map of V, we see tha the compactness condition (iii) is always satisfied. Therefore we can apply our theorem to this case (cf. [4]). 4. Spectral Embeddings and Spectral Convergence 4.1 We consider a nonnegative quadratic form ðE; D½EŠÞ on a set X satisfying the properties pffiffiffiffiffi (RF-i), (RF-ii), (RF-iv) and that REðx; yÞ > 0 for all x; y 2 X with x 6¼ y. We suppose that the metric space ðX; RE pffiffiffiffiffi Þ is separable. Let be a Borel measure on X ¼ðX; REÞ with support K such that Z ðXÞð¼ ðKÞÞ < þ1; REðx0; xÞ d ðxÞ < þ1 for some x0 2 X (and hence for any x0). Denote the cardinality of the set K by N ¼ #Kð þ1Þ. Letf i j 0 ¼ 0 < 1 2 ; 0 i N 1g the set of eigenvalues of the self-adjoint operator L associated with the induced form E K on L 2 ðK; Þ and ¼f i j 0 i N 1g an orthonormal complete system of eigenfunctions. Recall that the effective resistance RE is expressed on K as follows: REðx; yÞ ¼ XN 1 i¼1 iðxÞ pffiffiffiffi i X iðyÞ pffiffiffiffi i In what follows, we understand i= ffiffiffiffi p i ¼ 0 for i N if N is finite. Let ‘ 2 ¼fðaiÞ j P1 i¼1 a2i < þ1g, and define a map I of K into ‘2 by 2 ; x; y 2 K:
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Variational Convergence of Finite Networks

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