  We consider the quickest change-point detection problem where the aim is to
detect the onset of a pre-specified drift in "live"-monitored standard Brownian
motion; the change-point is assumed unknown (nonrandom). The topic of interest
is the distribution of the Generalized Shryaev-Roberts (GSR) detection
statistic set up to "sense" the presence of the drift. Specifically, we derive
a closed-form formula for the transition probability density function (pdf) of
the time-homogeneous Markov diffusion process generated by the GSR statistic
when the Brownian motion under surveillance is "drift-free", i.e., in the
pre-change regime; the GSR statistic's (deterministic) nonnegative headstart is
assumed arbitrarily given. The transition pdf formula is found analytically,
through direct solution of the respective Kolmogorov forward equation via the
Fourier spectral method to achieve separation of the spacial and temporal
variables. The obtained result generalizes the well-known formula for the
(pre-change) stationary distribution of the GSR statistic: the latter's
stationary distribution is the temporal limit of the distribution sought in
this work. To conclude, we exploit the obtained formula numerically and briefly
study the pre-change behavior of the GSR statistic versus three factors: (a)
drift-shift magnitude, (b) time, and (c) the GSR statistic's headstart.
