  Stochastic reaction networks are dynamical models of biochemical reaction
systems and form a particular class of continuous-time Markov chains on
$\mathbb{N}^n$. Here we provide a fundamental characterisation that connects
structural properties of a network to its dynamical features. Specifically, we
define the notion of `stochastically complex balanced systems' in terms of the
network's stationary distribution and provide a characterisation of
stochastically complex balanced systems, parallel to that established in the
70-80ies for deterministic reaction networks. Additionally, we establish that a
network is stochastically complex balanced if and only if an associated
deterministic network is complex balanced (in the deterministic sense), thereby
proving a strong link between the theory of stochastic and deterministic
networks. Further, we prove a stochastic version of the `deficiency zero
theorem' and show that any (not only complex balanced) deficiency zero reaction
network has a product-form Poisson-like stationary distribution on all
irreducible components.
  Finally, we provide sufficient conditions for when a product-form
Poisson-like distribution on a single (or all) component(s) implies the network
is complex balanced, and explore the possibility to characterise complex
balanced systems in terms of product-form Poisson-like stationary
distributions.
