  Recent experiments have probed the relative likelihoods of trajectories in
stochastic systems by observing survival probabilities within a tube of radius
$R$ in spacetime. We measure such probabilities here for a colloidal particle
in a corrugated channel, corresponding to a bistable potential with
state-dependent diffusivity. In contrast to previous findings for
state-independent noise, we find that the most probable pathway changes
qualitatively as the tube radius $R$ is altered. We explain this by computing
the survival probabilities predicted by overdamped Langevin dynamics. At high
enough resolution (small enough $R$), survival probabilities depend solely on
diffusivity variations, independent of deterministic forces; finite $R$
corrections yield a generalization of the Onsager-Machlup action. As corollary,
ratios of survival probabilities are singular as $R \to 0$, but become regular,
and described by the classical Onsager-Machlup action, only in the special case
of state-independent noise.
