  New time-dependent treatment of tunneling from localized state to continuum
is proposed. It does not use the Laplace transform (Green's function's method)
and can be applied for time-dependent potentials, as well. This approach
results in simple expressions describing dynamics of tunneling to Markovian and
non-Markovian reservoirs in the time-interval $-\infty<t<\infty$. It can
provide a new outlook for tunneling in the negative time region, illuminating
the origin of the time's arrow problem in quantum mechanics. We also
concentrate on singularity at $t=0$, which affects the perturbative expansion
of the evolution operator. In addition, the decay to continuum in periodically
modulated tunneling Hamiltonian is investigated. Using our results, we extend
the Tien-Gordon approach for periodically driven transport, to oscillating
tunneling barriers.
