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Forum Mathematicum

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Hrsg. v. Blomer, Valentin / Cohen, Frederick R. / Droste, Manfred / Duzaar, Frank / Echterhoff, Siegfried / Frahm, Jan / Gordina, Maria / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

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Band 28, Heft 5


Regular subspaces of skew product diffusions

Liping Li / Jiangang Ying
Online erschienen: 07.10.2015 | DOI: https://doi.org/10.1515/forum-2015-0012


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 the 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 E1×E2 and expressed as


where Xi is a symmetric diffusion on Ei for i=1,2, and A is a positive continuous additive functional of X1. One of our main results indicates that any skew product type regular subspace of X, say


can be characterized as follows: the associated smooth measure of A~ is equal to that of A, and Yi corresponds to a regular subspace of Xi for i=1,2. Furthermore, we shall make some discussions on rotationally invariant diffusions on d{}, which are special skew product diffusions on (0,)×Sd-1. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on d{} to a new regular Dirichlet form on d. More precisely, fix a regular Dirichlet form (,) on the state space d. Its part Dirichlet form on d{} is denoted by (0,)0. Let (~0,~)0 be a regular subspace of (0,)0. We want to find a regular subspace (~,~) of (,) such that the part Dirichlet form of (~,~) on d{} is exactly (~0,~)0. If (~,~) exists, we call it a regular extension of (~0,~)0. We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of (0,)0 has a unique regular extension.

Keywords: Regular subspaces; Dirichlet forms; skew product; rotation invariance

MSC 2010: 31C25; 60J55; 60J60


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Erhalten: 18.01.2015

Revidiert: 29.06.2015

Online erschienen: 07.10.2015

Erschienen im Druck: 01.09.2016

Funding Source: National Natural Science Foundation of China

Award identifier / Grant number: 11271240

Research supported in part by NSFC grant 11271240.

Quellenangabe: Forum Mathematicum, Band 28, Heft 5, Seiten 857–872, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2015-0012.

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