We construct an analogue of Dyson Brownian motion in the Siegel half-space \(\mathcal {H}\) that we term Siegel Brownian motion. Given \(\beta \in (0,\infty ]\) , a stochastic flow for \(Z_t\in \mathcal {H}\) is introduced so that the law of the eigenvalues \(\lambda _t\) of the cross ratio matrix \({\mathfrak {R}}(Z_t,\varvec{i}I_n)\) is determined, after a change of variables to \(\sigma (\lambda ) \in (0,\infty )^n\) , by the Itô differential equation 0.1 \(\begin{aligned} \textrm{d}\sigma ^k_t=\frac{1}{2}\left( \coth {\sigma ^k_t}+\sum _{l\ne k}\frac{\sinh {\sigma ^k_t}}{\cosh {\sigma ^k_t}-\cosh {\sigma ^l_t}}\right) \textrm{d}t+\sqrt{\frac{2}{\beta }}\textrm{d}W^k_t, \quad k=1,\ldots , n, \end{aligned}\) where \(W_t\) is a standard Wiener process in \(\mathbb {R}^n\) . This interacting particle system corresponds to stochastic gradient ascent 0.2 \(\begin{aligned} \textrm{d}\sigma _t= \frac{1}{2}\nabla S(\sigma _t) +\sqrt{\frac{2}{\beta }}\textrm{d}W_t, \end{aligned}\) where \(S(\sigma )= \log \textrm{vol}\,\mathcal {O}_{\lambda (\sigma )}\) is a Boltzmann entropy that enumerates the microstates in the group orbit \(\mathcal {O}_\lambda = \{Z \in \mathcal {H}\left| \textrm{eig}\left( {\mathfrak {R}}(Z,\varvec{i}I_n)\right) =\lambda \right. \}\) . In the limit \(\beta =\infty \) , the group orbits \(\mathcal {O}_{\lambda _t}\) evolve by motion by minus a half times mean curvature.
