# Convergence of the empirical spectral distribution of Gaussian matrix-valued processes

For a given normalized Gaussian symmetric matrix-valued process $Y^{(n)}=(Y^{(n)}(t); t\geq0)$, we consider the process of its eigenvalues $((\lambda_{1}^{(n)}(t),\dots, \lambda_{n}^{(n)}(t)); t\ge 0)$, and prove that, under some mild conditions on the covariance function associated to $Y^{(n)}$, the empirical spectral distribution converges in probability to a deterministic limit $(\mu_{t}, t\geq0)$, in the topology of weak convergence of probability measures; which is characterized by its Cauchy transform in terms of the solution of a Burgers' equation. Our results extend those of Pardo et al. for the non-commutative fractional Brownian motion when $H>1/2$, Rogers and Shi for the free Brownian motion $H=1/2$.