  As is well known the Complex Langevin (CL) method sometimes fails to converge
or converges to the wrong limit. We identified one reason for this long ago:
insufficient decay of the probability density either near infinity or near
poles of the drift, leading to boundary terms that spoil the formal argument
for correctness. To gain a deeper understanding of this phenomenon, we analyze
the emergence of such boundary terms thoroughly in a simple model, where
analytic results can be compared with numerics.
  We also show how some simple modification stabilizes the CL process in such a
way that it can produce results agreeing with direct integration. Besides
explicitly demonstrating the connection between boundary terms and correct
convergence our analysis also suggests a correctness criterion which could be
applied in realistic lattice simulations.
