  Consider the $n \times n$ reverse circulant $RC_n(t)$ and symmetric circulant
$SC_n(t)$ matrices with independent Brownian motion entries. We discuss the
process convergence of the time dependent fluctuations of linear eigenvalue
statistics of these matrices as $n \tends \infty$, when the test functions of
the statistics are polynomials. The proofs are mainly combinatorial, based on
the trace formula, method of moments and some results on process convergence.
