  In this paper we aim at characterizing the effect of stochastic fluctuations
on the distribution of the energy exchanged by a quantum system with an
external environment under sequences of quantum measurements performed at
random times. Both quenched and annealed averages are considered. The
information about fluctuations is encoded in the quantum-heat probability
density function, or equivalently in its characteristic function, whose general
expression for a quantum system with arbitrary Hamiltonian is derived. We prove
that, when a stochastic protocol of measurements is applied, the quantum
Jarzynski equality is obeyed. Therefore, the fluctuation relation is robust
against the presence of randomness in the times intervals between measurements.
Then, for the paradigmatic case of a two-level system, we analytically
characterize the quantum-heat transfer. Particular attention is devoted to the
limit of large number of measurements and to the effects caused by the
stochastic fluctuations. The relation with the stochastic Zeno regime is also
discussed.
