  Let (\rho_\la)_{\la\in \La} be a holomorphic family of representations of a
surface group \pi_1(S) into PSL(2,C), where S is a topological (possibly
punctured) surface with negative Euler characteristic. Given a structure of
Riemann surface of finite type on S we construct a bifurcation current on the
parameter space \La, that is a (1,1) positive closed current attached to the
bifurcations of the family. It is defined as the $dd^c$ of the Lyapunov
exponent of the representation with respect to the Brownian motion on the
Riemann surface S, endowed with its Poincare metric. We show that this
bifurcation current describes the asymptotic distribution of various
codimension 1 phenomena in \La. For instance, the random hypersurfaces of \La
defined by the condition that a random closed geodesic on S is mapped under
\rho_\la to a parabolic element or the identity are asymptotically
equidistributed with respect to the bifurcation current. The proofs are based
on our previous work "Random walks, Kleinian groups, and bifurcation currents",
and on a careful control of a discretization procedure of the Brownian motion.
