  For a conformally-coupled scalar field we obtain the conformally-related
Einstein-Langevin equations, using appropriate transformations for all the
quantities in the equations between two conformally-related spacetimes. In
particular, we analyze the transformations of the influence action, the stress
energy tensor, the noise kernel and the dissipation kernel. In due course the
fluctuation-dissipation relation is also discussed. The analysis in this paper
thereby facilitates a general solution to the Einstein-Langevin equation once
the solution of the equation in a simpler, conformally-related spacetime is
known. For example, from the Minkowski solution of Martin and Verdaguer, those
of the Einstein-Langevin equations in conformally-flat spacetimes, especially
for spatially-flat Friedmann-Robertson-Walker models, can be readily obtained.
