  Quantum Zeno Dynamics is the phenomenon that the observation or strong
driving of a quantum system can freeze its dynamics to a subspace, effectively
truncating the Hilbert space of the system. It represents the quantum version
of the famous flying arrow Zeno paradox. Here, we study how temporal
stochasticity in the system observation (or driving) affects the survival
probability of the system in the subspace. In particular, we introduce a strong
and a weak Zeno regime for which we quantify the confinement by providing an
analytical expression for this survival probability. We investigate several
dissipative and coherent protocols to confine the dynamics, and show that they
can be successfully adapted to the stochastic version. In the weak Zeno regime
the dynamics within the subspace effectively acts as an additional source of
stochasticity in the confinement protocol. Our analytical predictions are
numerically tested and verified on a paradigmatic spin chain. As practical
implications different coherent and dissipative confinement protocols allow to
choose a trade-off between a probabilistic scheme with high fidelity (compared
to perfect subspace dynamics) and a deterministic one with a slightly lower
fidelity, which is a step towards better control in future quantum
technologies.
