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:

$${\displaystyle d\lambda _{i}=dB_{i}+\sum _{1\leq j\leq n:j\neq i}{\frac {dt}{\lambda _{i}-\lambda _{j}}}}$$

${\displaystyle B_{1},...,B_{n}}$

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