  In this paper, we report a theoretical study of the phase diffusion in a
gain-switched single-mode semiconductor laser. We use stochastic rate equations
for the electrical field to analyze the phase statistics of the gain-switched
laser. Their use avoid the instabilities obtained with rate equations for
photon number and optical phase when the photon number is small. However we
show that a new problem appears when integrating with the field equations: the
variance of the optical phase becomes divergent. This divergence can not be
observed with the numerical integration of the commonly used equations for
photon number and optical phase because of the previous instabilities. The
divergence of the phase variance means that this quantity does not reach a
fixed value as the integration time step is decreased. We obtain that the phase
variance increases as the integration time step decreases with no sign of
saturation behaviour even for tiny steps. We explain the divergence by making
the analogy of our problem with the 2-dimensional Brownian motion. The fact
that the divergence appears is not surprising because already in 1940 Paul
L\`evy demonstrated that the variance of the polar angle in a 2-dimensional
Brownian motion is a divergent quantity. Our results show that stochastic rate
equations for photon number and phase are not appropriated for describing the
phase statistics when the photon number is small. Simulation of the stochastic
rate equations for the electrical field are consistent with L\`evy's results
but gives unphysical results since an infinite value is obtained for a quantity
that can be measured.
