  We present rigorous results for the mean first passage time and first passage
time statistics for two-channel Markov additive diffusion in a 3-dimensional
spherical domain. Inspired by biophysical examples we assume that the particle
can only recognise the target in one of the modes, which is shown to effect a
non-trivial first passage behaviour. We also address the scenario of
intermittent immobilisation. In both cases we prove that despite the perfectly
non-recurrent motion of two-channel Markov additive diffusion in 3 dimensions
the first passage statistics at long times do not display Poisson-like
behaviour if none of the phases has a vanishing diffusion coefficient. This
stands in stark contrast to the standard (one-channel) Markov diffusion
counterpart. We also discuss the relevance of our results in the context of
cellular signalling.
