This report deals with continuous limits of several one-dimensional diffusive systems, obtained from stochastic distortions of discrete curves with different kinds of coding. These systems are indeed special cases of reaction-diffusion. A general functional formalism is set up, allowing to grapple with hydrodynamic limits. We also analyse the steady-state regime, not only in the reversible case, so that the invariant measure can have a non Gibbs form. A link is made between recursion properties, which originate matrix solutions, and particle cycles in the state-graph, by introducing loop currents on the analogy with electric circuits. Also, by means of the aforementioned functional approach, a bridge is established between structural constants involved in the recursions at discrete level and the constants which appear in Lotka-Volterra equations describing the fluid limits of stationary states. Finally the Lagrangian for the current fluctuations is obtained from an iterative scheme, and the related Hamilton-Jacobi equation, leading to the large deviation functional, is solved at least in the reversible case allowing to rediscover some known results.
