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:$$d \lambda_i=d B_i+\sum_{1 \leq j \leq n: j \neq i} \frac{d t}{\lambda_i-\lambda_j}$$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 $\lambda_1(0), \lambda_2(0), ..., \lambda_n(0)$.