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:[math]\displaystyle{ d \lambda_i=d B_i+\sum_{1 \leq j \leq n: j \neq i} \frac{d t}{\lambda_i-\lambda_j} }[/math]where [math]\displaystyle{ B_1, ..., B_n }[/math] are different and independent Wiener processes.

Start with a Hermitian matrix with eigenvalues [math]\displaystyle{ \lambda_1(0), \lambda_2(0), ..., \lambda_n(0) }[/math], then let it perform Brownian motion in the space of Hermitian matrices. Its eigenvalues constitute a Dyson Brownian motion.

Start with [math]\displaystyle{ n }[/math] independent Wiener processes started at different locations [math]\displaystyle{ \lambda_1(0), \lambda_2(0), ..., \lambda_n(0) }[/math], then condition on those processes to be non-intersecting for all time. The resulting process is a Dyson Brownian motion starting at the same [math]\displaystyle{ \lambda_1(0), \lambda_2(0), ..., \lambda_n(0) }[/math].^[4]

References

↑ Dyson, Freeman J. (1962-11-01). "A Brownian-Motion Model for the Eigenvalues of a Random Matrix" (in en). Journal of Mathematical Physics 3 (6): 1191–1198. doi:10.1063/1.1703862. ISSN 0022-2488. https://pubs.aip.org/jmp/article/3/6/1191/228277/A-Brownian-Motion-Model-for-the-Eigenvalues-of-a.
↑ 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, https://www.cambridge.org/core/books/first-course-in-random-matrix-theory/dyson-brownian-motion/F63EE7DFFF72FE3FD0A1B5B5F42818A2, retrieved 2023-11-25
↑ Tao, Terence (2010-01-19). "254A, Notes 3b: Brownian motion and Dyson Brownian motion" (in en). https://terrytao.wordpress.com/2010/01/18/254a-notes-3b-brownian-motion-and-dyson-brownian-motion/.
↑ Grabiner, David J. (1999). "Brownian motion in a Weyl chamber, non-colliding particles, and random matrices" (in en). Annales de l'I.H.P. Probabilités et statistiques 35 (2): 177–204. ISSN 1778-7017. http://www.numdam.org/item/?id=AIHPB_1999__35_2_177_0.

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