We study the problem of identification of a proper state-space for the
stochastic dynamics of free particles in continuum, with their possible birth
and death. In this dynamics, the motion of each separate particle is described
by a fixed Markov process M on a Riemannian manifold X. The main problem
arising here is a possible collapse of the system, in the sense that, though
the initial configuration of particles is locally finite, there could exist a
compact set in X such that, with probability one, infinitely many particles
will arrive at this set at some time t>0. We assume that X has infinite
volume and, for each α≥1, we consider the set Θα of all
infinite configurations in X for which the number of particles in a compact
set is bounded by a constant times the α-th power of the volume of the
set. We find quite general conditions on the process M which guarantee that
the corresponding infinite particle process can start at each configuration
from Θα, will never leave Θα, and has cadlag (or,
even, continuous) sample paths in the vague topology. We consider the following
examples of applications of our results: Brownian motion on the configuration
space, free Glauber dynamics on the configuration space (or a birth-and-death
process in X), and free Kawasaki dynamics on the configuration space. We also
show that if X=Rd, then for a wide class of starting distributions,
the (non-equilibrium) free Glauber dynamics is a scaling limit of
(non-equilibrium) free Kawasaki dynamics