On unique extension of time changed reflecting Brownian motions

Zhen Qing Chen; Masatoshi Fukushima

doi:10.1214/08-AIHP301

On unique extension of time changed reflecting Brownian motions

Zhen Qing Chen^*, Masatoshi Fukushima

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

9 Citations (Scopus)

Abstract

Let D be an unbounded domain in ℝ^d with d ≥ 3. We show that if D contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on D is transient. Next assume that RBM X on D is transient and let Y be its time change by Revuz measure 1_D(x)m(x) dx for a strictly positive continuous integrable function m on D. We further show that if there is some r > 0 so that D \ B(0, r) is an unbounded uniform domain, then Y admits one and only one symmetric diffusion that genuinely extends it and admits no killings.

Original language	English
Pages (from-to)	864-875
Number of pages	12
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	45
Issue number	3
DOIs	https://doi.org/10.1214/08-AIHP301
Publication status	Published - Aug 2009
Externally published	Yes

Keywords

BL function space
Diffusion extension
Harmonic function
Reflected Dirichlet space
Reflecting Brownian motion
Sobolev space
Time change
Transience
Uniform domain

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abstract = "Let D be an unbounded domain in ℝd with d ≥ 3. We show that if D contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on D is transient. Next assume that RBM X on D is transient and let Y be its time change by Revuz measure 1D(x)m(x) dx for a strictly positive continuous integrable function m on D. We further show that if there is some r > 0 so that D \ B(0, r) is an unbounded uniform domain, then Y admits one and only one symmetric diffusion that genuinely extends it and admits no killings.",

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AB - Let D be an unbounded domain in ℝd with d ≥ 3. We show that if D contains an unbounded uniform domain, then the symmetric reflecting Brownian motion (RBM) on D is transient. Next assume that RBM X on D is transient and let Y be its time change by Revuz measure 1D(x)m(x) dx for a strictly positive continuous integrable function m on D. We further show that if there is some r > 0 so that D \ B(0, r) is an unbounded uniform domain, then Y admits one and only one symmetric diffusion that genuinely extends it and admits no killings.

KW - BL function space

KW - Diffusion extension

KW - Harmonic function

KW - Reflected Dirichlet space

KW - Reflecting Brownian motion

KW - Sobolev space

KW - Time change

KW - Transience

KW - Uniform domain

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JF - Annales de l'institut Henri Poincare (B) Probability and Statistics

IS - 3

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On unique extension of time changed reflecting Brownian motions

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Keywords

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