  A central challenge in physics is to describe non-equilibrium systems driven
by randomness, such as a randomly growing interface, or fluids subject to
random fluctuations that account e.g. for local stresses and heat fluxes not
related to the velocity and temperature gradients. For deterministic systems
with infinitely many degrees of freedom, normal form and center manifold theory
have shown a prodigious efficiency to often completely characterize how the
onset of linear instability translates into the emergence of nonlinear
patterns. However, in presence of random fluctuations, this reduction procedure
is seriously challenged due to large excursions caused by the noise, and the
approach needs to be revisited.
  We present an alternative framework to cope with these difficulties
exploiting the approximation theory of stochastic invariant manifolds and
energy estimates measuring the defect of parameterization of the high-modes. To
operate for fluid problems, these error estimates are derived under assumptions
regarding dissipation effects brought by the high-modes that suitably
counterbalance the loss of regularity due to the nonlinear terms. The approach
enables us to predict, from the reduced equations, the occurrence in large
probability of a stochastic analogue to the pitchfork bifurcation, as long as
the noise's intensity and the eigenvalue's magnitude of the mildly unstable
mode scale accordingly.
  Our parameterization formulas involve non-Markovian coefficients, which
depend explicitly on the history of the noise path that drives the SPDE
dynamics, and their memory content is self-consistently determined by the
intensity of the random force and its interaction through the SPDE's nonlinear
terms. Applications to a stochastic Rayleigh-Benard problem are detailed, for
which conditions for a stochastic pitchfork bifurcation (in large probability)
to occur, are clarified.
