Equivalent single junction model for two-junction quantum interferometers with small inductance in the presence of noise

We illustrate, in the limit of small values of the parameter $\beta$, a method to analyze the stochastic equations governing the time evolution of a two-junction quantum interferometer in the presence of noise. The analysis takes into account the different characteristic times of the two processes involved in the system, in such a way that the appropriate probability density function for each process is derived by means of one-dimensional Fokker-Planck equations. Therefore, the present analysis is equivalent to a generalization of the reduced single-junction model in the case of thermal fluctuations in the junctions. Voltage-current characteristics, transfer functions and circulating currents in 0-SQUIDs and $\pi$-SQUIDs are calculated by means of the proposed analytic approach