One-point reflection

Zhen Qing Chen; Masatoshi Fukushima

doi:10.1016/j.spa.2014.11.002

One-point reflection

Zhen Qing Chen^*, Masatoshi Fukushima

^*Corresponding author for this work

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

16 Citations (Scopus)

Abstract

We examine symmetric extensions of symmetric Markov processes with one boundary point. Relationship among various normalizations of local times, entrance laws and excursion laws is studied. Dirichlet form characterization of elastic one-point reflection of symmetric Markov processes is derived. We give a direct construction of Walsh's Brownian motion as a one-point reflection together with its Dirichlet form characterization. This yields directly the analytic characterization of harmonic and subharmonic functions for Walsh's Brownian motion, recently obtained by Fitzsimmons and Kuter (2014) using a different method. We further study as a one-point reflection two-dimensional Brownian motion with darning (BMD).

Original language	English
Pages (from-to)	1368-1393
Number of pages	26
Journal	Stochastic Processes and their Applications
Volume	125
Issue number	4
DOIs	https://doi.org/10.1016/j.spa.2014.11.002
Publication status	Published - Apr 2015
Externally published	Yes

Keywords

Boundary theory
Brownian motion with darning
Conformal invariance
Dirichlet form
Excursion law
Harmonic functions
Local time
One-point reflection

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10.1016/j.spa.2014.11.002

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title = "One-point reflection",

abstract = "We examine symmetric extensions of symmetric Markov processes with one boundary point. Relationship among various normalizations of local times, entrance laws and excursion laws is studied. Dirichlet form characterization of elastic one-point reflection of symmetric Markov processes is derived. We give a direct construction of Walsh's Brownian motion as a one-point reflection together with its Dirichlet form characterization. This yields directly the analytic characterization of harmonic and subharmonic functions for Walsh's Brownian motion, recently obtained by Fitzsimmons and Kuter (2014) using a different method. We further study as a one-point reflection two-dimensional Brownian motion with darning (BMD).",

keywords = "Boundary theory, Brownian motion with darning, Conformal invariance, Dirichlet form, Excursion law, Harmonic functions, Local time, One-point reflection",

author = "Chen, {Zhen Qing} and Masatoshi Fukushima",

year = "2015",

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language = "English",

volume = "125",

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journal = "Stochastic Processes and their Applications",

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T1 - One-point reflection

AU - Chen, Zhen Qing

AU - Fukushima, Masatoshi

PY - 2015/4

Y1 - 2015/4

N2 - We examine symmetric extensions of symmetric Markov processes with one boundary point. Relationship among various normalizations of local times, entrance laws and excursion laws is studied. Dirichlet form characterization of elastic one-point reflection of symmetric Markov processes is derived. We give a direct construction of Walsh's Brownian motion as a one-point reflection together with its Dirichlet form characterization. This yields directly the analytic characterization of harmonic and subharmonic functions for Walsh's Brownian motion, recently obtained by Fitzsimmons and Kuter (2014) using a different method. We further study as a one-point reflection two-dimensional Brownian motion with darning (BMD).

AB - We examine symmetric extensions of symmetric Markov processes with one boundary point. Relationship among various normalizations of local times, entrance laws and excursion laws is studied. Dirichlet form characterization of elastic one-point reflection of symmetric Markov processes is derived. We give a direct construction of Walsh's Brownian motion as a one-point reflection together with its Dirichlet form characterization. This yields directly the analytic characterization of harmonic and subharmonic functions for Walsh's Brownian motion, recently obtained by Fitzsimmons and Kuter (2014) using a different method. We further study as a one-point reflection two-dimensional Brownian motion with darning (BMD).

KW - Boundary theory

KW - Brownian motion with darning

KW - Conformal invariance

KW - Dirichlet form

KW - Excursion law

KW - Harmonic functions

KW - Local time

KW - One-point reflection

UR - http://www.scopus.com/inward/record.url?scp=84922570904&partnerID=8YFLogxK

U2 - 10.1016/j.spa.2014.11.002

DO - 10.1016/j.spa.2014.11.002

M3 - Article

AN - SCOPUS:84922570904

SN - 0304-4149

VL - 125

SP - 1368

EP - 1393

JO - Stochastic Processes and their Applications

JF - Stochastic Processes and their Applications

IS - 4

ER -

One-point reflection

Abstract

Keywords

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