  We construct global-in-time singular dynamics for the (renormalized) cubic
fourth order nonlinear Schr\"odinger equation on the circle, having the white
noise measure as an invariant measure. For this purpose, we introduce the
"random-resonant / nonlinear decomposition", which allows us to single out the
singular component of the solution. Unlike the classical McKean, Bourgain, Da
Prato-Debussche type argument, this singular component is nonlinear, consisting
of arbitrarily high powers of the random initial data. We also employ a random
gauge transform, leading to random Fourier restriction norm spaces. For this
problem, a contraction argument does not work and we instead establish
convergence of smooth approximating solutions by studying the partially
iterated Duhamel formulation under the random gauge transform. We reduce the
crucial nonlinear estimates to boundedness properties of certain random
multilinear functionals of the white noise.
