On eigenvalues of the Brownian sheet matrix

Brownian sheet; Dyson Brownian motion; Empirical spectral measure; High-dimensional limit; McKean–Vlasov equation; Random matrix; Dyson brownian motion; Eigen-value; High-dimensional; Higher-dimensional; McKean-Vlasov equations; Random Matrix; Spectral measure; Statistics and Probability; Modeling and Simulation; Applied Mathematics

Abstract :

[en] We derive a system of stochastic partial differential equations satisfied by the eigenvalues of the symmetric matrix whose entries are the Brownian sheets. We prove that the sequence Ld(s,t),(s,t)∈[0,S]×[0,T]d∈N of empirical spectral measures of the rescaled matrices is tight on C([0,S]×[0,T],P(R)) and hence is convergent as d goes to infinity by Wigner's semicircle law. We also obtain PDEs which are satisfied by the high-dimensional limiting measure.

Disciplines :

Mathematics

Author, co-author :

Song, Jian ; Research Center for Mathematics and Interdisciplinary Sciences, Shandong University, Shandong, China

Xiao, Yimin; Department of Statistics and Probability, Michigan State University, East Lansing, United States

YUAN, Wangjun ; University of Luxembourg > Faculty of Science, Technology and Medicine (FSTM) > Department of Mathematics (DMATH)

External co-authors :

yes

Language :

English

Title :

On eigenvalues of the Brownian sheet matrix

Publication date :

December 2023

Journal title :

Stochastic Processes and Their Applications

ISSN :

0304-4149

eISSN :

1879-209X

Publisher :

Elsevier B.V.

Volume :

166

Pages :

104231

Peer reviewed :

Peer Reviewed verified by ORBi

Additional URL :

https://api.elsevier.com/content/article/PII:S030441492300203X?httpAccept=text/xml

Funders :

National Science Foundation
Engineering Research Centers
National Natural Science Foundation of China
Natural Science Foundation of Shandong Province

Funding text :

We are deeply grateful to an anonymous reviewer for his/her comments which have helped us to improve the presentation of our results. J. Song is partially supported by National Natural Science Foundation of China (nos. 12071256 and 12226001 ) and Major Basic Research Program of the Natural Science Foundation of Shandong Province in China (nos. ZR2019ZD42 and ZR2020ZD24 ). Y. Xiao is partially supported by grants (nos. DMS-1855185 and DMS-2153846 ) from the National Science Foundation of the U.S.A . Wangjun Yuan is partially supported by ERC Consolidator Grant (no. 815703 ) “STAMFORD: Statistical Methods for High Dimensional Diffusions”.

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since 27 November 2023

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