  A system under constant observation is practically freezed to the measurement
subspace. If the system driving is a random classical field, the survival
probability of the system in the subspace becomes a random variable described
by the Stochastic Quantum Zeno Dynamics (SQZD) formalism. Here, we study the
time and ensemble average of this random survival probability and demonstrate
how time correlations in the noisy environment determine whether the two
averages do coincide or not. These environment time correlations can
potentially generate non-Markovian dynamics of the quantum system depending on
the structure and energy scale of the system Hamiltonian. We thus propose a way
to probe this interesting property of the environment by means of the system
survival probability. This will further contribute to the development of new
schemes for quantum sensing technologies, where nanodevices may be exploited to
image external structures or biological molecules via the surface field they
generate.
