  Recently two generalized nonlinear Schr\"{o}dinger equations have been
proposed by Chavanis [Eur. Phys. J. Plus 132 (2017) 286] by applying Nottale's
theory of scale relativity relying on a fractal space-time to describe
dissipation in quantum systems. Several existing nonlinear equations are then
derived and discussed in this context leading to a continuity equation with an
extra source/sink term which violates Ehrenfest theorem. An extension to
describe stochastic dynamics is also carried out by including thermal
fluctuations or noise of the environment. These two generalized nonlinear
equations are analyzed within the Bohmian mechanics framework to describe the
corresponding dissipative and stochastic dynamics in terms of quantum
trajectories. Several applications of this second generalized equation which
can be considered as a generalized Kostin equation have been carried out. The
first application consists of the study of the position-momentum uncertainty
principle in a dissiaptive dynamics. After, the so-called Brownian-Bohmian
motion is investigated by calculating classical and quantum diffusion
coefficients. And as a third example, transmission through a transient (time
dependent) parabolic repeller is studied where the interesting phenomenon of
early arrival is observed even in the stochastic dynamics although the
magnitude of early arrival is reduced by friction.
