  We describe Stochastic Loewner Evolution on arbitrary Riemann surfaces with
boundary using Conformal Field Theory methods. We propose in particular a CFT
construction for a probability measure on (clouded) paths, and check it against
known restriction properties. The probability measure can be thought of as a
section of the determinant bundle over moduli spaces of Riemann surfaces.
Loewner evolutions have a natural description in terms of random walk in the
moduli space, and the stochastic diffusion equation translates to the Virasoro
action of a certain weight-two operator on a uniformised version of the
determinant bundle.
