Dyson Brownian motion

In mathematics, the Dyson Brownian motion is a real-valued continuous-time stochastic process named for Freeman Dyson.^[1] Dyson studied this process in the context of random matrix theory.

There are several equivalent definitions:^[2]^[3]

Definition by stochastic differential equation: $d\lambda _{i}=dB_{i}+\sum _{1\leq j\leq n:j\neq i}{\frac {dt}{\lambda _{i}-\lambda _{j}}}$ where $B_{1},...,B_{n}$ are different and independent Wiener processes.

Start with a Hermitian matrix with eigenvalues ${\textstyle \lambda _{1}(0),\lambda _{2}(0),...,\lambda _{n}(0)}$ , then let it perform Brownian motion in the space of Hermitian matrices. Its eigenvalues constitute a Dyson Brownian motion.

Start with ${\textstyle n}$ independent Wiener processes started at different locations ${\textstyle \lambda _{1}(0),\lambda _{2}(0),...,\lambda _{n}(0)}$ , then condition on those processes to be non-intersecting for all time. The resulting process is a Dyson Brownian motion starting at the same ${\textstyle \lambda _{1}(0),\lambda _{2}(0),...,\lambda _{n}(0)}$ .^[4]

References

[edit]

^ Dyson, Freeman J. (1962-11-01). "A Brownian-Motion Model for the Eigenvalues of a Random Matrix". Journal of Mathematical Physics. 3 (6): 1191–1198. doi:10.1063/1.1703862. ISSN 0022-2488.
^ Bouchaud, Jean-Philippe; Potters, Marc, eds. (2020), "Dyson Brownian Motion", A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists, Cambridge: Cambridge University Press, pp. 121–135, ISBN 978-1-108-48808-2, retrieved 2023-11-25
^ Tao, Terence (2010-01-19). "254A, Notes 3b: Brownian motion and Dyson Brownian motion". What's new. Retrieved 2023-11-25.
^ Grabiner, David J. (1999). "Brownian motion in a Weyl chamber, non-colliding particles, and random matrices". Annales de l'I.H.P. Probabilités et statistiques. 35 (2): 177–204. ISSN 1778-7017.

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