Regular subspaces of skew product diffusions

Liping Li; Jiangang Ying

doi:10.1515/forum-2015-0012

Article

Regular subspaces of skew product diffusions

Liping Li and Jiangang Ying

Published/Copyright: October 7, 2015

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From the journal Forum Mathematicum Volume 28 Issue 5

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 E1×E2 and expressed as

Xt=(Xt1,XAt2),t≥0,

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

Yt=(Yt1,YA~t2),t≥0,

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 ℝd∖{}, which are special skew product diffusions on (0,∞)×Sd-1. Our main purpose is to extend a regular subspace of rotationally invariant diffusion on ℝd∖{} to a new regular Dirichlet form on ℝd. More precisely, fix a regular Dirichlet form (ℰ,ℱ) on the state space ℝd. Its part Dirichlet form on ℝd∖{} is denoted by (ℰ0,ℱ)0. Let (ℰ~0,ℱ~)0 be a regular subspace of (ℰ0,ℱ)0. We want to find a regular subspace (ℰ~,ℱ~) of (ℰ,ℱ) such that the part Dirichlet form of (ℰ~,ℱ~) on ℝd∖{} is exactly (ℰ~0,ℱ~)0. If (ℰ~,ℱ~) exists, we call it a regular extension of (ℰ~0,ℱ~)0. We shall prove that, under a mild assumption, any rotationally invariant type regular subspace of (ℰ0,ℱ)0 has a unique regular extension.

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

MSC 2010: 31C25; 60J55; 60J60

Communicated by Ichiro Shigekawa

Funding source: National Natural Science Foundation of China

Award Identifier / Grant number: 11271240

Funding statement: Research supported in part by NSFC grant 11271240.

This work was initiated when the first author visited the University of California, San Diego. He would like to thank Professor Patrick J. Fitzsimmons for his hospitality and many helpful discussions. We also want to thank the anonymous reviewers for pointing out the article [21] that we missed before.

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Received: 2015-1-18

Revised: 2015-6-29

Published Online: 2015-10-7

Published in Print: 2016-9-1

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Articles in the same Issue

https://doi.org/10.1515/forum-2015-0012

Keywords for this article

Regular subspaces; Dirichlet forms; skew product; rotation invariance