I think the classical Poissonian process is where you have something, which in a time dt has a probability ωdt. Then one can show quite easily that the probability that the "something" has not yet decayed goes as P(t)=exp(-ωt), because it obeys a differential equation with the given solution. However, what does P(t) look like if ω is time dependent?
Just like before, you have to solve the differential equation [itex]P'(t) = -\omega(t)P(t)[/itex]. The general solution is [itex]P(t) = \exp\left(-\int_0^t \omega(u)\,du\right).[/itex]