  We define a new diffusive matrix model converging towards the $\beta$ -Dyson
Brownian motion for all $\beta\in [0,2]$ that provides an explicit construction
of $\beta$-ensembles of random matrices that is invariant under the
orthogonal/unitary group. For small values of $\beta$, our process allows one
to interpolate smoothly between the Gaussian distribution and the Wigner
semi-circle. The interpolating limit distributions form a one parameter family
that can be explicitly computed. This also allows us to compute the finite-size
corrections to the semi-circle.
