  We investigate the asymptotic properties of Markov processes associated to
stochastic chemical reaction networks (CRNs) driven by the kinetics of the law
of mass action. Their transition rates exhibit a polynomial dependence on the
state variable, with possible discontinuities of the dynamics along the
boundary of the state space. As a natural choice to study stability properties
of CRNs, the scaling parameter considered in this paper is the norm of the
initial state. Compared to existing scalings of the literature, this scaling
does not change neither the topology of a CRN, nor its reactions constants.
Functional limit theorems with this scaling parameter can be used to prove
positive recurrence of the Markov process. This scaling approach also gives
interesting insights on the transient behavior of these networks, to describe
how multiple time scales drive the time evolution of their sample paths for
example. General stability criteria are presented as well as a possible
framework for scaling analyses. Several simple examples of CRNs are
investigated with this approach. A detailed stability and scaling analyses of a
CRN with slow and fast timescales is worked out.

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