Language evolution in a noisy environment

We study language formation through reinforcement learning in a signaling game over a noisy channel. We show that under the general assumption of memory-less channel, many of the results that hold for similar dynamics in a noiseless environment, hold in the presence of a noisy channel. In particular, we show that conditioned on the existence of the solution for the fluid dynamics, such dynamics admits a weak Lyapunov function. Furthermore, we show that original dynamics is convergent to the zero set of the derivative of this Lyapunov function along the trajectories of the dynamics.