What Is a Stochastic Hamiltonian Process on Finite Graph? An Optimal Transport Answer

Jianbo Cui; Shu Liu; Haomin Zhou

doi:10.1016/j.jde.2021.10.009

What Is a Stochastic Hamiltonian Process on Finite Graph? An Optimal Transport Answer

Jianbo Cui
, Shu Liu
, Haomin Zhou

Research output: Journal article publication › Journal article › Academic research › peer-review

8 Citations (Scopus)

Abstract

We present a definition of stochastic Hamiltonian process on finite graph via its corresponding density dynamics in Wasserstein manifold. We demonstrate the existence of stochastic Hamiltonian process in many classical discrete problems, such as the optimal transport problem, Schrödinger equation and Schrödinger bridge problem (SBP). The stationary and periodic properties of Hamiltonian processes are also investigated in the framework of SBP.

Original language	English
Pages (from-to)	428-457
Number of pages	30
Journal	Journal of Differential Equations
Volume	305
DOIs	https://doi.org/10.1016/j.jde.2021.10.009
Publication status	Published - 25 Dec 2021

Keywords

Optimal transport
Schrödinger bridge problem
Time-inhomogeneous Markov process
Wasserstein-Hamiltonian flow

ASJC Scopus subject areas

Analysis
Applied Mathematics

More information

10.1016/j.jde.2021.10.009

http://hdl.handle.net/10397/91600

Cite this

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title = "What Is a Stochastic Hamiltonian Process on Finite Graph? An Optimal Transport Answer",

abstract = "We present a definition of stochastic Hamiltonian process on finite graph via its corresponding density dynamics in Wasserstein manifold. We demonstrate the existence of stochastic Hamiltonian process in many classical discrete problems, such as the optimal transport problem, Schr{\"o}dinger equation and Schr{\"o}dinger bridge problem (SBP). The stationary and periodic properties of Hamiltonian processes are also investigated in the framework of SBP.",

keywords = "Optimal transport, Schr{\"o}dinger bridge problem, Time-inhomogeneous Markov process, Wasserstein-Hamiltonian flow",

author = "Jianbo Cui and Shu Liu and Haomin Zhou",

note = "Funding Information: The authors are very grateful to Prof. Shui-Nee Chow for introducing this problem and his helpful suggestions. The research is partially supported by Georgia Tech Mathematics Application Portal (GT-MAP) and by research grants NSF DMS-1620345 , DMS-1830225 , and ONR N00014-18-1-2852 . Publisher Copyright: {\textcopyright} 2021 Elsevier Inc.",

year = "2021",

month = dec,

day = "25",

doi = "10.1016/j.jde.2021.10.009",

language = "English",

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journal = "Journal of Differential Equations",

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T1 - What Is a Stochastic Hamiltonian Process on Finite Graph? An Optimal Transport Answer

AU - Cui, Jianbo

AU - Liu, Shu

AU - Zhou, Haomin

N1 - Funding Information: The authors are very grateful to Prof. Shui-Nee Chow for introducing this problem and his helpful suggestions. The research is partially supported by Georgia Tech Mathematics Application Portal (GT-MAP) and by research grants NSF DMS-1620345 , DMS-1830225 , and ONR N00014-18-1-2852 . Publisher Copyright: © 2021 Elsevier Inc.

PY - 2021/12/25

Y1 - 2021/12/25

N2 - We present a definition of stochastic Hamiltonian process on finite graph via its corresponding density dynamics in Wasserstein manifold. We demonstrate the existence of stochastic Hamiltonian process in many classical discrete problems, such as the optimal transport problem, Schrödinger equation and Schrödinger bridge problem (SBP). The stationary and periodic properties of Hamiltonian processes are also investigated in the framework of SBP.

AB - We present a definition of stochastic Hamiltonian process on finite graph via its corresponding density dynamics in Wasserstein manifold. We demonstrate the existence of stochastic Hamiltonian process in many classical discrete problems, such as the optimal transport problem, Schrödinger equation and Schrödinger bridge problem (SBP). The stationary and periodic properties of Hamiltonian processes are also investigated in the framework of SBP.

KW - Optimal transport

KW - Schrödinger bridge problem

KW - Time-inhomogeneous Markov process

KW - Wasserstein-Hamiltonian flow

UR - https://www.scopus.com/pages/publications/85117700758

U2 - 10.1016/j.jde.2021.10.009

DO - 10.1016/j.jde.2021.10.009

M3 - Journal article

AN - SCOPUS:85117700758

SN - 0022-0396

VL - 305

SP - 428

EP - 457

JO - Journal of Differential Equations

JF - Journal of Differential Equations

ER -

What Is a Stochastic Hamiltonian Process on Finite Graph? An Optimal Transport Answer

Abstract

Keywords

ASJC Scopus subject areas

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