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:$${\displaystyle d\lambda _{i}=dB_{i}+\sum _{1\leq j\leq n:j\neq i}{\frac {dt}{\lambda _{i}-\lambda _{j}}}}$$where ${\displaystyle B_{1},...,B_{n}}$ 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. This is defined within the Weyl chamber ${\displaystyle W_{n}:=\{(x_{1},\dots ,x_{n})\in \mathbb {R} ^{n}:x_{1}<\dots <x_{n}\}}$, as well as any coordinate-permutation of it.

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 ${\textstyle \lambda _{1}(0),\lambda _{2}(0),...,\lambda _{n}(0)}$.

In Random Matrix Theory, the Gaussian unitary ensemble is a fundamental ensemble. It is defined as a probability distribution over the space of ${\textstyle A\in \mathbb {R} ^{n\times n}}$ Hermitian matrices, with probability density function ${\displaystyle \rho (A)\propto e^{-{\frac {1}{2}}tr(A^{2})}}$.

Consider a Hermitian matrix ${\textstyle A\in \mathbb {R} ^{n\times n}}$. The space of Hermitian matrices can be mapped to the space of real vectors ${\textstyle \mathbb {R} ^{n^{2}}}$: $${\displaystyle A\mapsto (A_{11},\dots ,A_{nn},{\sqrt {2}}Re(A_{12}),\dots ,{\sqrt {2}}Re(A_{n-1,n}),{\sqrt {2}}Im(A_{12}),\dots ,{\sqrt {2}}Im(A_{n-1,n}))}$$This is an isometry, where the matrix norm is Frobenius norm. By reversing this process, a standard Brownian motion in ${\textstyle \mathbb {R} ^{n^{2}}}$ maps back to a Brownian motion in the space of ${\textstyle n\times n}$ Hermitian matrices:$${\displaystyle dA={\begin{bmatrix}dB_{11}&{\frac {1}{\sqrt {2}}}(dB_{12}+idB_{12}')&{\frac {1}{\sqrt {2}}}(dB_{13}+idB_{13}')&\cdots &{\frac {1}{\sqrt {2}}}(dB_{1n}+idB_{1n}')\\{\frac {1}{\sqrt {2}}}(dB_{12}-idB_{12}')&dB_{22}&{\frac {1}{\sqrt {2}}}(dB_{23}+idB_{23}')&\cdots &{\frac {1}{\sqrt {2}}}(dB_{2n}+idB_{2n}')\\{\frac {1}{\sqrt {2}}}(dB_{13}-idB_{13}')&{\frac {1}{\sqrt {2}}}(dB_{23}-idB_{23}')&dB_{33}&\cdots &{\frac {1}{\sqrt {2}}}(dB_{3n}+idB_{3n}')\\\vdots &\vdots &\vdots &\ddots &\vdots \\{\frac {1}{\sqrt {2}}}(dB_{1n}-idB_{1n}')&{\frac {1}{\sqrt {2}}}(dB_{2n}-idB_{2n}')&{\frac {1}{\sqrt {2}}}(dB_{3n}-idB_{3n}')&\cdots &dB_{nn}\end{bmatrix}}}$$The claim is that the eigenvalues of ${\displaystyle A}$ evolve according to$${\displaystyle d\lambda _{i}=dB_{i}+\sum _{1\leq j\leq n:j\neq i}{\frac {dt}{\lambda _{i}-\lambda _{j}}}}$$

Since each ${\textstyle dB}$ is on the order of ${\textstyle O({\sqrt {dt}})}$, we can equivalently write ${\textstyle dA={\sqrt {dt}}G}$, where ${\textstyle G}$ is a random Hermitian matrix where each entry is on the order of ${\textstyle O(1)}$. By construction of the standard Brownian motion, ${\textstyle dA}$ is independent of ${\textstyle A}$, so ${\textstyle G}$ is independent of ${\textstyle A}$, and can be written as $${\displaystyle dA={\begin{bmatrix}X_{11}&{\frac {1}{\sqrt {2}}}(X_{12}+iY_{12})&{\frac {1}{\sqrt {2}}}(X_{13}+iY_{13})&\cdots &{\frac {1}{\sqrt {2}}}(X_{1n}+iY_{1n})\\{\frac {1}{\sqrt {2}}}(X_{12}-iY_{12})&X_{22}&{\frac {1}{\sqrt {2}}}(X_{23}+iY_{23})&\cdots &{\frac {1}{\sqrt {2}}}(X_{2n}+iY_{2n})\\{\frac {1}{\sqrt {2}}}(X_{13}-iY_{13})&{\frac {1}{\sqrt {2}}}(X_{23}-iY_{23})&X_{33}&\cdots &{\frac {1}{\sqrt {2}}}(X_{3n}+iY_{3n})\\\vdots &\vdots &\vdots &\ddots &\vdots \\{\frac {1}{\sqrt {2}}}(X_{1n}-iY_{1n})&{\frac {1}{\sqrt {2}}}(X_{2n}-iY_{2n})&{\frac {1}{\sqrt {2}}}(X_{3n}-iY_{3n})&\cdots &X_{nn}\end{bmatrix}}}$$ where each random variable ${\textstyle X_{ij},Y_{ij}}$ is standard normal. In other words, ${\textstyle G}$ is distributed according to the GUE(n).

By the first and second Hadamard variation formulas and Ito’s lemma, we have $${\displaystyle \lambda _{i}(t+dt)=\lambda _{i}(t)+{\sqrt {dt}}u_{i}^{*}Gu_{i}+dt\sum _{j\neq i}{\frac {\mathbb {E} [|u_{i}^{*}Gu_{j}|^{2}]}{\lambda _{i}-\lambda _{j}}}}$$