  We propose a method to sample stationary properties of solutions of
stochastic differential equations, which is accurate and efficient if there are
rarely visited regions or rare transitions between distinct regions of the
state space. The method is based on a complete, non-overlapping partition of
the state space into patches on which the stochastic process is ergodic. On
each of these patches we run simulations of the process strictly truncated to
the corresponding patch, which allows effective simulations also in rarely
visited regions. The correct weight for each patch is obtained by counting the
attempted transitions between all different patches. The results are
patchworked to cover the whole state space. We extend the concept of truncated
Markov chains which is originally formulated for processes which obey detailed
balance to processes not fulfilling detailed balance. The method is illustrated
by three examples, describing the one-dimensional diffusion of an overdamped
particle in a double-well potential, a system of many globally coupled
overdamped particles in double-well potentials subject to additive Gaussian
white noise, and the overdamped motion of a particle on the circle in a
periodic potential subject to a deterministic drift and additive noise. In the
appendix we explain how other well-known Markov chain Monte Carlo algorithms
can be related to truncated Markov chains.
