Probabilistic model of a crystal oscillator at low drive levels

The probability density of the envelope of crystal oscillator oscillations is studied at low drives when self-excitation becomes not available due to the large and nonlinear friction of a piezoelectric resonator that occurs after a long storage. We show that the effect know as "sleeping sickness" is accompanied with the noise-induced oscillations at a new steady state far from the equilibrium. The stochastic differential equation is given for the envelope of the oscillations induced by the interplay of the nonlinear resonator friction and multiplicative noise. The Fokker-Planck equation is discussed in a sense of Stratonovich. The stationary probability density of the envelope is derived and investigated