On the local property for positivity preserving coercive forms
On the local property for positivity preserving coercive forms
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论保存积极性的强制形式的局部属性
DOI：
10.1515/9783110880052.345
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发表时间：
1993
影响因子：
0.3
通讯作者：
B. Schmuland
中科院分区：
文献类型：
--
作者：
B. Schmuland
我们表明，在温和的条件下，两个著名的定义的局部性质的狄利克雷形式是等价的。我们还表明，来自微分算子的形式是本地的。1991 AMS Subject Classification：31C25本文的目的是澄清在狄利克雷形式文献中出现的两种不同局部性概念之间的关系。第一个是对Bouleau和Hirsch的书[BH 91; Chapter I，Corollary 5.1.4]中的定域性定义稍作修改的版本，而第二个来自Ma和Röckner的书[MR 92; Chapter V，Proposition 1.2]。但这里我们并不假设形式满足任何正规收缩性质，而只是假设它是保正的（见下面的定义0.1）。设（E，F，m）是测度空间，（E，D（E））是L2（E，F，m）上的一个稠密闭双线性型.根据[MR 92]，我们称这样的形式（E，D（E））为强制的，如果对所有u ∈ D（E），E（u，u）≥ 0，且对某些K ≥ 1，我们有，|E（u，v）|≤ KE1（u，u）E1（v，v）对于所有u，v ∈ D（E），（0.1）其中我们定义E1（u，v）：= E（u，v）+（u，v）L2（E; m）。定义0.1.以下是形式（E，D（E））可以满足的各种压缩性质：（C1）如果u ∈ D（E），则u + ∈ D（E）且E（u+，u −）≤ 0。(C2)如果u ∈ D（E），则u <$1 ∈ D（E）且E（u <$1，u − u <$1）≥ 0。(C3)如果u ∈ D（E），则u <$1 ∈ D（E）且E（u − u <$1，u <$1）≥ 0。条件（C1）等价于相关半群（Tt）t ≥ 0的正性（因此对偶半群（T t）t ≥ 0）;也就是说，对于u ∈ L2，如果0 ≤ u，则0 ≤ Ttu。因此，满足（C1）的形式（E，D（E））称为正性保持。条件（C2）等价于（Tt）t ≥ 0的马尔可夫性质;即，对于u ∈ L2，如果0 ≤ u ≤ 1，则0 ≤ Ttu ≤ 1，而条件（C3）等价于对偶半群（T_t）t ≥ 0的马尔可夫性质。我们注意到，（C2）或（C3）中的任何一个蕴涵（C1）。A表格（E，D（E））
We show that, under mild conditions, two well-known definitions for the local property of a Dirichlet form are equivalent. We also show that forms that come from differential operators are local. 1991 AMS Subject Classification: 31C25 The purpose of this paper is to clarify the relationship between two different notions of locality that have appeared in the literature of Dirichlet forms. The first is a slightly modified version of the definition of locality found in the book of Bouleau and Hirsch [BH 91; Chapter I, Corollary 5.1.4], while the second comes from the book of Ma and Röckner [MR 92; Chapter V, Proposition 1.2]. But here we do not assume that the form satisfies any normal contraction property, but only that it is positivity preserving (see Definition 0.1 below). Let (E,F ,m) be a measure space, and suppose (E , D(E)) is a densely defined, closed, bilinear form on L2(E,F ,m). Following [MR 92], we call such a form (E , D(E)) coercive if E(u, u) ≥ 0 for all u ∈ D(E), and for some K ≥ 1 we have, |E(u, v)| ≤ KE1(u, u)E1(v, v) for all u, v ∈ D(E), (0.1) where we define E1(u, v) := E(u, v) + (u, v)L2(E;m). Definition 0.1. The following are various contraction properties that the form (E , D(E)) may satisfy: (C1) If u ∈ D(E), then u+ ∈ D(E) and E(u+, u−) ≤ 0. (C2) If u ∈ D(E), then u ∧ 1 ∈ D(E) and E(u ∧ 1, u− u ∧ 1) ≥ 0. (C3) If u ∈ D(E), then u ∧ 1 ∈ D(E) and E(u− u ∧ 1, u ∧ 1) ≥ 0. Condition (C1) is equivalent with the positivity of the associated semigroup (Tt)t≥0, (and hence also the dual semigroup (T̂t)t≥0); that is, for u ∈ L2 if 0 ≤ u, then 0 ≤ Ttu. Hence, a form (E , D(E)) satisfying (C1) is called positivity preserving. Condition (C2) is equivalent to the Markov property for (Tt)t≥0; that is, for u ∈ L2 if 0 ≤ u ≤ 1, then 0 ≤ Ttu ≤ 1, while condition (C3) is equivalent to the Markov property for the dual semigroup (T̂t)t≥0. We note that either of (C2) or (C3) implies (C1). A form (E , D(E))