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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

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

## 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 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 ${E}_{1}×{E}_{2}$ and expressed as

${X}_{t}=\left({X}_{t}^{1},{X}_{{A}_{t}}^{2}\right),t\ge 0,$

where ${X}^{i}$ 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, say

${Y}_{t}=\left({Y}_{t}^{1},{Y}_{{\stackrel{~}{A}}_{t}}^{2}\right),t\ge 0,$

can be characterized as follows: the associated smooth measure of $\stackrel{~}{A}$ 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 ${ℝ}^{d}\setminus \left\{\right\}$, which are special skew product diffusions on $\left(0,\mathrm{\infty }\right)×{S}^{d-1}$. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on ${ℝ}^{d}\setminus \left\{\right\}$ to a new regular Dirichlet form on ${ℝ}^{d}$. More precisely, fix a regular Dirichlet form $\left(\mathcal{ℰ},\mathcal{ℱ}\right)$ on the state space ${ℝ}^{d}$. Its part Dirichlet form on ${ℝ}^{d}\setminus \left\{\right\}$ is denoted by $\left({\mathcal{ℰ}}^{0},\mathcal{ℱ}{}^{0}\right)$. Let $\left({\stackrel{~}{\mathcal{ℰ}}}^{0},\stackrel{~}{\mathcal{ℱ}}{}^{0}\right)$ be a regular subspace of $\left({\mathcal{ℰ}}^{0},\mathcal{ℱ}{}^{0}\right)$. We want to find a regular subspace $\left(\stackrel{~}{\mathcal{ℰ}},\stackrel{~}{\mathcal{ℱ}}\right)$ of $\left(\mathcal{ℰ},\mathcal{ℱ}\right)$ such that the part Dirichlet form of $\left(\stackrel{~}{\mathcal{ℰ}},\stackrel{~}{\mathcal{ℱ}}\right)$ on ${ℝ}^{d}\setminus \left\{\right\}$ is exactly $\left({\stackrel{~}{\mathcal{ℰ}}}^{0},\stackrel{~}{\mathcal{ℱ}}{}^{0}\right)$. If $\left(\stackrel{~}{\mathcal{ℰ}},\stackrel{~}{\mathcal{ℱ}}\right)$ exists, we call it a regular extension of $\left({\stackrel{~}{\mathcal{ℰ}}}^{0},\stackrel{~}{\mathcal{ℱ}}{}^{0}\right)$. We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of $\left({\mathcal{ℰ}}^{0},\mathcal{ℱ}{}^{0}\right)$ has a unique regular extension.

MSC 2010: 31C25; 60J55; 60J60

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## Artikelinformationen

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,

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