  The probability current statistics of two-dimensional open chaotic ballistic
billiards is studied both analytically and numerically. Assuming that the real
and imaginary parts of the scattering wave function are both random Gaussian
fields, we find a universal distribution function for the probability current.
In by-passing we recover previous analytic forms for wave function statistics.
The expressions bridge the entire region from GOE to GUE type statistics. Our
analytic expressions are verified numerically by explicit quantum-mechanical
calculations of transport through a Bunimovich billiard.
