  This study deals with continuous limits of interacting one-dimensional
diffusive systems, arising from stochastic distortions of discrete curves with
various kinds of coding representations. These systems are essentially of a
reaction-diffusion nature. In the non-reversible case, the invariant measure
has generally a non Gibbs form. The corresponding steady-state regime is
analyzed in detail with the help of a tagged particle and a state-graph cycle
expansion of the probability currents. As a consequence, the constants
appearing in Lotka-Volterraequations --which describe the fluid limits of
stationary states-- can be traced back directly at the discrete level to tagged
particles cycles coefficients. Current fluctuations are also studied and the
Lagrangian is obtained by an iterative scheme. The related Hamilton-Jacobi
equation, which leads to the large deviation functional, is analyzed and solved
in the reversible case for the sake of checking.
