  We present a stochastic approach for the description of the quantum dynamics
of open system in a fermionic environment (bath). The full quantum evolution as
provided by the Schrodinger equation is reformulated exactly as a probabilistic
average over the so-called dressed quantum trajectories. The latter are defined
as follows. The fermionic environment can be represented as a fermi sea whose
"surface" is covered by the ripples of quantum fluctuations. If we consider
these fluctuations in the basis of the particle-hole coherent states, then
these fluctuations produce a classical particle-hole noise.The probability
distribution of this noise is provided by the generalized particle-hole Husimi
function of the vacuum. Then we define the dressed quantum trajectory as the
evolution of the open system and the bath which is conditioned on a particular
particle-hole noise sample. The resulting description resembles the
non-Markovian quantum state diffusion for the bosonic bath. Therefore, we
expect that our fermionic approach will share its favourable propeties like the
possibility to carry out Monte-Carlo simulations of non-Markovian quantum
dynamics on long times.
