Dyson Brownian motion

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

There are several equivalent definitions:

Definition by stochastic differential equation:$$d{\lambda }_i=d{B}_i+\sum _{1\le j\le n:j\ne i}\frac{dt}{{\lambda }_i-{\lambda }_j}$$where $B_1,...,B_n$ are different and independent Wiener processes.

Start with a Hermitian matrix with eigenvalues ${\lambda }_1(0),{\lambda }_2(0),...,{\lambda }_n(0)$, then let it perform Brownian motion in the space of Hermitian matrices. Its eigenvalues constitute a Dyson Brownian motion.

Start with $n$ independent Wiener processes started at different locations ${\lambda }_1(0),{\lambda }_2(0),...,{\lambda }_n(0)$, then condition on those processes to be non-intersecting for all time. The resulting process is a Dyson Brownian motion starting at the same ${\lambda }_1(0),{\lambda }_2(0),...,{\lambda }_n(0)$.