On the Quantity of Information Processed by a Nonlinear System with Internal Noise

We investigate a random signal $x(t)$ passing through a system such that at the output there is obtained a process $y(t)$ related to $x(t)$ by a differential equation. The differential operator contains a term which depends on the internal noise of the system. It is assumed that this noise is Gaussian white noise, and the differential operator is a general nonlinear operator. Under the assumption that $x(t)$ is a Gaussian process, an expression is found for the quantity of information in the process $y(t)$ relative to $x(t)$. For the case when the relation between $x(t)$ and $y(t)$ is stationary, a rather simple expression for the rate of transmission is found.