  A series of recent articles introduced a method to construct stochastic
partial differential equations (SPDEs) which are invariant with respect to the
distribution of a given conditioned diffusion. These works are restricted to
the case of elliptic diffusions where the drift has a gradient structure and
the resulting SPDE is of second-order parabolic type. The present article
extends this methodology to allow the construction of SPDEs which are invariant
with respect to the distribution of a class of hypoelliptic diffusion
processes, subject to a bridge conditioning, leading to SPDEs which are of
fourth-order parabolic type. This allows the treatment of more realistic
physical models, for example, one can use the resulting SPDE to study
transitions between meta-stable states in mechanical systems with friction and
noise. In this situation the restriction of the drift being a gradient can also
be lifted.
