  We construct $P(phi)_1$-processes indexed by the full time-line, separately
derived from the functional integral representations of the relativistic and
non-relativistic Nelson models in quantum field theory. These two cases differ
essentially by sample path regularity. Associated with these processes we
define a martingale which, under an appropriate scaling, allows to obtain a
central limit theorem for additive functionals of these processes. We discuss a
number of examples by choosing specific functionals related to particle-field
operators.
