We present in this paper a differential version of Mirzakhani's recursion
relation for the Weil-Petersson volumes of the moduli spaces of bordered
Riemann surfaces. We discover that the differential relation, which is
equivalent to the original integral formula of Mirzakhani, is a Virasoro
constraint condition on a generating function for these volumes. We also show
that the generating function for psi and kappa_1 intersections on the moduli
space of stable algebraic curves is a 1-parameter solution to the KdV
hierarchy. It recovers the Witten-Kontsevich generating function when the
parameter is set to be 0.Comment: 21 pages, 3 figures; v3. new introduction, minor revision
