  We develop a formalism to calculate the quasi-particle energy within the GW
many-body perturbation correction to the density functional theory (DFT). The
occupied and virtual orbitals of the Kohn-Sham (KS) Hamiltonian are replaced by
stochastic orbitals used to evaluate the Green function, the polarization
potential, and thereby the GW self-energy. The stochastic GW (sGW) relies on
novel theoretical concepts such as stochastic time-dependent Hartree
propagation, stochastic matrix compression and spatial/temporal stochastic
decoupling techniques. Beyond the theoretical interest, the formalism enables
linear scaling GW calculations breaking the theoretical scaling limit for GW as
well as circumventing the need for energy cutoff approximations. We illustrate
the method for silicon nanocrystals of varying sizes with over 3000 electrons.
