  We present a statistical physics framework for description of nonlinear
non-equilibrium stochastic processes, modeled via chemical master equation, in
the weak-noise limit. Using the Poisson representation approach and applying
the large-deviation principle we first solve the master equation. Then we use
the notion of the non-equilibrium free energy to derive an integral fluctuation
relation for nonlinear non-equilibrium systems under feedback control. We point
out that the free energy as well as some functionals can serve as
non-equilibrium Lyapunov function which has an important property to decay to
its minimal value monotonously at all times. The Poisson representation
technique is illustrated via exact stochastic treatment of biophysical
processes, such as bacterial chemosensing and molecular evolution.
