Infinite dimensional stochastic differential equations for Dyson’s model

Li Cheng Tsai

doi:10.1007/s00440-015-0672-2

Infinite dimensional stochastic differential equations for Dyson’s model

Li Cheng Tsai

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

17 Scopus citations

Abstract

In this paper we show the strong existence and the pathwise uniqueness of an infinite-dimensional stochastic differential equation (SDE) corresponding to the bulk limit of Dyson’s Brownian Motion, for all β≥ 1. Our construction applies to an explicit and general class of initial conditions, including the lattice configuration { x_i} = Z and the sine process. We further show the convergence of the finite to infinite-dimensional SDE. This convergence concludes the determinantal formula of Katori and Tanemura (Commun Math Phys 293(2):469–497, 2010) for the solution of this SDE at β= 2.

Original language	English (US)
Pages (from-to)	801-850
Number of pages	50
Journal	Probability Theory and Related Fields
Volume	166
Issue number	3-4
DOIs	https://doi.org/10.1007/s00440-015-0672-2
State	Published - Dec 1 2016
Externally published	Yes

All Science Journal Classification (ASJC) codes

Analysis
Statistics and Probability
Statistics, Probability and Uncertainty

Keywords

Correlation function
Dyson’s Brownian motion
Dyson’s model
Infinite-dimensional
Pathwise uniqueness
Stochastic differential equations
Strong existence

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10.1007/s00440-015-0672-2

Cite this

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title = "Infinite dimensional stochastic differential equations for Dyson{\textquoteright}s model",

abstract = "In this paper we show the strong existence and the pathwise uniqueness of an infinite-dimensional stochastic differential equation (SDE) corresponding to the bulk limit of Dyson{\textquoteright}s Brownian Motion, for all β≥ 1. Our construction applies to an explicit and general class of initial conditions, including the lattice configuration { xi} = Z and the sine process. We further show the convergence of the finite to infinite-dimensional SDE. This convergence concludes the determinantal formula of Katori and Tanemura (Commun Math Phys 293(2):469–497, 2010) for the solution of this SDE at β= 2.",

keywords = "Correlation function, Dyson{\textquoteright}s Brownian motion, Dyson{\textquoteright}s model, Infinite-dimensional, Pathwise uniqueness, Stochastic differential equations, Strong existence",

author = "Tsai, {Li Cheng}",

note = "Funding Information: LCT thanks Alexei Borodin for suggesting this direction of research, Amir Dembo for many fruitful discussions, and the anonymous reviewers for improving the presentation of this paper. LCT is partially supported by the NSF through DMS-0709248. Publisher Copyright: {\textcopyright} 2015, Springer-Verlag Berlin Heidelberg.",

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doi = "10.1007/s00440-015-0672-2",

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T1 - Infinite dimensional stochastic differential equations for Dyson’s model

AU - Tsai, Li Cheng

N1 - Funding Information: LCT thanks Alexei Borodin for suggesting this direction of research, Amir Dembo for many fruitful discussions, and the anonymous reviewers for improving the presentation of this paper. LCT is partially supported by the NSF through DMS-0709248. Publisher Copyright: © 2015, Springer-Verlag Berlin Heidelberg.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - In this paper we show the strong existence and the pathwise uniqueness of an infinite-dimensional stochastic differential equation (SDE) corresponding to the bulk limit of Dyson’s Brownian Motion, for all β≥ 1. Our construction applies to an explicit and general class of initial conditions, including the lattice configuration { xi} = Z and the sine process. We further show the convergence of the finite to infinite-dimensional SDE. This convergence concludes the determinantal formula of Katori and Tanemura (Commun Math Phys 293(2):469–497, 2010) for the solution of this SDE at β= 2.

AB - In this paper we show the strong existence and the pathwise uniqueness of an infinite-dimensional stochastic differential equation (SDE) corresponding to the bulk limit of Dyson’s Brownian Motion, for all β≥ 1. Our construction applies to an explicit and general class of initial conditions, including the lattice configuration { xi} = Z and the sine process. We further show the convergence of the finite to infinite-dimensional SDE. This convergence concludes the determinantal formula of Katori and Tanemura (Commun Math Phys 293(2):469–497, 2010) for the solution of this SDE at β= 2.

KW - Correlation function

KW - Dyson’s Brownian motion

KW - Dyson’s model

KW - Infinite-dimensional

KW - Pathwise uniqueness

KW - Stochastic differential equations

KW - Strong existence

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JO - Probability Theory and Related Fields

JF - Probability Theory and Related Fields

IS - 3-4

ER -

Infinite dimensional stochastic differential equations for Dyson’s model

Abstract

All Science Journal Classification (ASJC) codes

Keywords

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