  We study the escape rate of a particle in a metastable potential in presence
of a dissipative bath coupled to the momentum of the particle. Using the
semiclassical bounce technique, we find that this rate is exponentially
enhanced. In particular, the influence of momentum dissipation depends on the
slope of the barrier that the particle is tunneling through. We investigate
also the influence of dissipative baths coupled to the position, and to the
momentum of the particle, respectively. In this case the rate exhibits a
non-monotonic behavior as a function of the dissipative coupling strengths.
Remarkably, even in presence of position dissipation, momentum dissipation can
enhance exponentially the escape rate in a large range of the parameter space.
The influence of the momentum dissipation is also witnessed by the substantial
increase of the average energy loss during inelastic (environment-assisted)
tunneling.
