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. 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.
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