Uniqueness for reflecting Brownian motion in lip domains

Richard F. Bass; Krzysztof Burdzy; Zhen Qing Chen

doi:10.1016/j.anihpb.2004.06.001

Uniqueness for reflecting Brownian motion in lip domains

Richard F. Bass, Krzysztof Burdzy^*, Zhen Qing Chen

^*Corresponding author for this work

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

19 Citations (Scopus)

Abstract

A lip domain is a Lipschitz domain where the Lipschitz constant is strictly less than one. We prove strong existence and pathwise uniqueness for the solution X = {X_t, t ≥ 0} to the Skorokhod equation dX_t = dW_t + n(X_t)dL_t, in planar lip domains, where W = {W_t, t ≥ 0 } is a Brownian motion, n is the inward pointing unit normal vector, and L = {L_t, t ≥ 0} is a local time on the boundary which satisfies some additional regularity conditions. Counterexamples are given for some Lipschitz (but not lip) three dimensional domains.

Original language	English
Pages (from-to)	197-235
Number of pages	39
Journal	Annales de l'institut Henri Poincare (B) Probability and Statistics
Volume	41
Issue number	2
DOIs	https://doi.org/10.1016/j.anihpb.2004.06.001
Publication status	Published - Mar 2005
Externally published	Yes

Keywords

Lip domain
Lipschitz domain
Local time
Pathwise uniqueness
Reflecting Brownian motion
Skorokhod equation
Strong existence
Weak uniqueness

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10.1016/j.anihpb.2004.06.001

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@article{78222c39f31446e69408c30b99405172,

title = "Uniqueness for reflecting Brownian motion in lip domains",

abstract = "A lip domain is a Lipschitz domain where the Lipschitz constant is strictly less than one. We prove strong existence and pathwise uniqueness for the solution X = {Xt, t ≥ 0} to the Skorokhod equation dXt = dWt + n(Xt)dLt, in planar lip domains, where W = {Wt, t ≥ 0 } is a Brownian motion, n is the inward pointing unit normal vector, and L = {Lt, t ≥ 0} is a local time on the boundary which satisfies some additional regularity conditions. Counterexamples are given for some Lipschitz (but not lip) three dimensional domains.",

keywords = "Lip domain, Lipschitz domain, Local time, Pathwise uniqueness, Reflecting Brownian motion, Skorokhod equation, Strong existence, Weak uniqueness",

author = "Bass, {Richard F.} and Krzysztof Burdzy and Chen, {Zhen Qing}",

year = "2005",

month = mar,

doi = "10.1016/j.anihpb.2004.06.001",

language = "English",

volume = "41",

pages = "197--235",

journal = "Annales de l'institut Henri Poincare (B) Probability and Statistics",

issn = "0246-0203",

publisher = "Institute of Mathematical Statistics",

number = "2",

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TY - JOUR

T1 - Uniqueness for reflecting Brownian motion in lip domains

AU - Bass, Richard F.

AU - Burdzy, Krzysztof

AU - Chen, Zhen Qing

PY - 2005/3

Y1 - 2005/3

N2 - A lip domain is a Lipschitz domain where the Lipschitz constant is strictly less than one. We prove strong existence and pathwise uniqueness for the solution X = {Xt, t ≥ 0} to the Skorokhod equation dXt = dWt + n(Xt)dLt, in planar lip domains, where W = {Wt, t ≥ 0 } is a Brownian motion, n is the inward pointing unit normal vector, and L = {Lt, t ≥ 0} is a local time on the boundary which satisfies some additional regularity conditions. Counterexamples are given for some Lipschitz (but not lip) three dimensional domains.

AB - A lip domain is a Lipschitz domain where the Lipschitz constant is strictly less than one. We prove strong existence and pathwise uniqueness for the solution X = {Xt, t ≥ 0} to the Skorokhod equation dXt = dWt + n(Xt)dLt, in planar lip domains, where W = {Wt, t ≥ 0 } is a Brownian motion, n is the inward pointing unit normal vector, and L = {Lt, t ≥ 0} is a local time on the boundary which satisfies some additional regularity conditions. Counterexamples are given for some Lipschitz (but not lip) three dimensional domains.

KW - Lip domain

KW - Lipschitz domain

KW - Local time

KW - Pathwise uniqueness

KW - Reflecting Brownian motion

KW - Skorokhod equation

KW - Strong existence

KW - Weak uniqueness

UR - http://www.scopus.com/inward/record.url?scp=14544299191&partnerID=8YFLogxK

U2 - 10.1016/j.anihpb.2004.06.001

DO - 10.1016/j.anihpb.2004.06.001

M3 - Article

AN - SCOPUS:14544299191

SN - 0246-0203

VL - 41

SP - 197

EP - 235

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

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

IS - 2

ER -

Uniqueness for reflecting Brownian motion in lip domains

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Keywords

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