  In many applications, it is important to be able to sample paths of SDEs
conditional on observations of various kinds. This paper studies SPDEs which
solve such sampling problems. The SPDE may be viewed as an infinite-dimensional
analogue of the Langevin equation used in finite-dimensional sampling. In this
paper, conditioned nonlinear SDEs, leading to nonlinear SPDEs for the sampling,
are studied. In addition, a class of preconditioned SPDEs is studied, found by
applying a Green's operator to the SPDE in such a way that the invariant
measure remains unchanged; such infinite dimensional evolution equations are
important for the development of practical algorithms for sampling infinite
dimensional problems. The resulting SPDEs provide several significant
challenges in the theory of SPDEs. The two primary ones are the presence of
nonlinear boundary conditions, involving first order derivatives, and a loss of
the smoothing property in the case of the pre-conditioned SPDEs. These
challenges are overcome and a theory of existence, uniqueness and ergodicity is
developed in sufficient generality to subsume the sampling problems of interest
to us. The Gaussian theory developed in Part I of this paper considers Gaussian
SDEs, leading to linear Gaussian SPDEs for sampling. This Gaussian theory is
used as the basis for deriving nonlinear SPDEs which affect the desired
sampling in the nonlinear case, via a change of measure.
