  We prove for an infinite array of globally coupled overdamped anharmonic
oscillators subject to additive Gaussian white noise the existence of a
well-behaved critical manifold in the parameter space which separates a
symmetric phase from a symmetry broken phase. Given two of the system
parameters there is an unique critical value of the third. The proof exploits
that the critical control parameter a_c is bounded by its limit values for weak
and for strong noise. In these limits the mechanism of symmetry breaking
differs. For weak noise the distribution is Gaussian and the symmetry is broken
as the whole distribution is shifted in either the positive or the negative
direction. For strong noise there is a symmetric double-peak distribution and
the symmetry is broken as the weights of the peaks become different. We derive
an ordinary differential equation whose solution describes the critical
manifold. Using a series ansatz to solve this differential equation we
determine the critical manifold for weak and for strong noise and compare it to
numerical results. We derive analytic expressions for the order parameter and
the susceptibility close to the critical manifold.
