  The sojourn probability of an It\^o diffusion process, i.e. its probability
to remain in the tubular neighborhood of a smooth path, is a central quantity
in the study of path probabilities. For $N$-dimensional It\^o processes with
state-dependent full-rank diffusion tensor, we derive a general expression for
the sojourn probability in tubes whose radii are small but finite, and fixed by
the metric of the ambient Euclidean space. The central quantity in our study is
the exit rate at which trajectories leave the tube for the first time. This has
an interpretation as a Lagrangian and can be measured directly in experiment,
unlike previously defined sojourn probabilities which depend on prior knowledge
of the state-dependent diffusivity. We find that while in the limit of
vanishing tube radius the ratio of sojourn probabilities for a pair of distinct
paths is in general divergent, the same for a path and its time-reversal is
always convergent and finite. This provides a pathwise definition of
irreversibility for It\^o processes that is agnostic to the state-dependence of
the diffusivity. For one-dimensional systems we derive an explicit expression
for our Lagrangian in terms of the drift and diffusivity, and find that our
result differs from previously reported multiplicative-noise Lagrangians. We
confirm our result by comparing to numerical simulations, and relate our theory
to the Stratonovich Lagrangian for multiplicative noise. For one-dimensional
systems, we discuss under which conditions the vanishing-radius limiting ratio
of sojourn probabilities for a pair of forward and backward paths recovers the
established pathwise entropy production. Finally, we demonstrate for our
one-dimensional example system that the most probable tube for a barrier
crossing depends sensitively on the tube radius, and hence on the tolerated
amount of fluctuations around the smooth reference path.
