  We consider the interplay between nonlocal nonlinearity and randomness for
two different nonlinear Schr\"odinger models. We show that stability of bright
solitons in presence of random perturbations increases dramatically with the
nonlocality-induced finite correlation length of the noise in the transverse
plane, by means of both numerical simulations and analytical estimates. In
fact, solitons are practically insensitive to noise when the correlation length
of the noise becomes comparable to the extent of the wave packet. We
characterize soliton stability using two different criteria based on the
evolution of the Hamiltonian of the soliton and its power. The first criterion
allows us to estimate a time (or distance) over which the soliton preserves its
form. The second criterion gives the life-time of the solitary wave packet in
terms of its radiative power losses. We derive a simplified mean field approach
which allows us to calculate the power loss analytically in the physically
relevant case of weakly correlated noise, which in turn serves as a lower
estimate of the life-time for correlated noise in general case.
