Dirichlet forms associated with linear diffusions

Fang, Xing; He, Ping; Ying, Jia

doi:10.1007/s11401-010-0589-0

Cited by 22 publications

(34 citation statements)

References 11 publications

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“…An irreducible one-dimensional diffusion must be symmetric with respect to the speed measure and its Dirichlet form has representation given in [4,Theorem 3.1]. Then applying [4,Theorem 4.1], we conclude that if an (irreducible) regular Dirichlet form is a Dirichlet extension of one-dimensional Brownian motion if and only if the scale function of its associated diffusion belongs to the following class: Note that T(R) = {t = s −1 |s ∈ S(R)} since the range s(R) of s may be a proper subset of R for some s ∈ S(R) (such as the example at the end of [3]). At least we have proper examples, such as Example 3.16, for the problem (Q.1).…”

Section: Introductionmentioning

confidence: 99%

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Regular Dirichlet extensions of one-dimensional Brownian motion

Li¹,

Ying²

2019

Ann. Inst. H. Poincaré Probab. Statist.

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The regular Dirichlet extension is the dual concept of regular Dirichlet subspace. The main purpose of this paper is to characterize all the regular Dirichlet extensions of onedimensional Brownian motion and to explore their structures. It is shown that every regular Dirichlet extension of one-dimensional Brownian motion may essentially decomposed into at most countable disjoint invariant intervals and an E-polar set relative to this regular Dirichlet extension. On each invariant interval the regular Dirichlet extension is characterized uniquely by a scale function in a given class. To explore the structure of regular Dirichlet extension we apply the idea introduced in [17], we formulate the trace Dirichlet forms and attain the darning process associated with the restriction to each invariant interval of the orthogonal complement of H 1 e (R) in the extended Dirichlet space of the regular Dirichlet extension. As a result, we find an answer to a long-standing problem whether a pure jump Dirichlet form has proper regular Dirichlet subspaces.

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Section: Introductionmentioning

confidence: 99%

“…The existence of t will be explained later. Referring to [4], the Dirichlet form of X on L 2 (R) is given by…”

Section: Introductionmentioning

confidence: 99%

Regular Dirichlet extensions of one-dimensional Brownian motion

Li¹,

Ying²

2019

Ann. Inst. H. Poincaré Probab. Statist.

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“…It indicates that we can reduce the problems about regular subspaces of rotationally invariant diffusions to those of their radius parts, i.e. minimal diffusions on (0, ∞), which have been considered in [6]. The next remark contains some general conclusions about global properties of regular subspaces, which can be found in [5] [19] and [20].…”

Section: Corollary 33 Let the Notations And Conditions Be The Same mentioning

confidence: 97%

“…, where r ′ is a diffusion on (0, ∞) with speed measure x d−1 dx and scale function p ′ such that p ′ is absolutely continuous with respect to p, dp ′ dp = 0 or 1, a.e., (Cf. [6] [19] and [20]), and…”

Section: Corollary 33 Let the Notations And Conditions Be The Same mentioning

confidence: 99%

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Regular subspaces of skew product diffusions

Ying

2015

Forum Mathematicum

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Abstract. Roughly speaking, the regular subspace of a Dirichlet form is also a regular Dirichlet form on the same state space. It inherits the same form of original Dirichlet form but possesses a smaller domain. What we are concerned in this paper are the regular subspaces of associated Dirichlet forms of skew product diffusions. A skew product diffusion X is a symmetric Markov process on the product state space E 1 × E 2 and expressed aswhere X i is a symmetric diffusion on E i for i = 1, 2, and A is a positive continuous additive functional of X 1 . One of our main results indicates that any skew product type regular subspace of X, saycan be characterized as follows: the associated smooth measure ofÃ is equal to that of A, and Y i corresponds to a regular subspace of X i for i = 1, 2. Furthermore, we shall make some discussions on rotationally invariant diffusions on R d \ {0}, which are special skew product diffusions on (0, ∞) × S d−1 . Our main purpose is to extend a regular subspace of rotationally invariant diffusion on R d \ {0} to a new regular Dirichlet form on R d . More precisely, fix a regular Dirichlet form (E, F ) on the state space R d . Its part Dirichlet form on R d \ {0} is denoted by (E 0 , F 0 ). Let (Ẽ 0 ,F 0 ) be a regular subspace of (E 0 , F 0 ). We want to find a regular subspace (Ẽ,F ) of (E, F ) such that the part Dirichlet form of (Ẽ,F ) on R d \ {0} is exactly (Ẽ 0 ,F 0 ). If (Ẽ ,F ) exists, we call it a regular extension of (Ẽ 0 ,F 0 ). We shall prove that under a mild assumption, any rotationally invariant type regular subspace of (E 0 , F 0 ) has a unique regular extension.

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Silverstein Extension and Fukushima Extension

Ying

2022

Springer Proceedings in Mathematics &Amp; Statistics

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Dirichlet forms associated with linear diffusions

Cited by 22 publications

References 11 publications

Regular Dirichlet extensions of one-dimensional Brownian motion

Regular Dirichlet extensions of one-dimensional Brownian motion

Regular subspaces of skew product diffusions

Silverstein Extension and Fukushima Extension

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