  A constant-rate multi-mode system is a hybrid system that can switch freely
among a finite set of modes, and whose dynamics is specified by a finite number
of real-valued variables with mode-dependent constant rates. We introduce and
study a stochastic extension of a constant-rate multi-mode system where the
dynamics is specified by mode-dependent compactly supported probability
distributions over a set of constant rate vectors. Given a tolerance
$\varepsilon > 0$, the almost-sure reachability problem for stochastic
multi-mode systems is to decide the existence of a control strategy that steers
the system almost-surely from an arbitrary start state to an
$\varepsilon$-neighborhood of an arbitrary target state while staying inside a
pre-specified safety set. We prove a necessary and sufficient condition to
decide almost-sure reachability and, using this condition, we show that
almost-sure reachability can be decided in polynomial time. Our algorithm can
be used as a path-following algorithm in combination with any off-the-shelf
path-planning algorithm to make a robot or an autonomous vehicle with noisy
low-level controllers follow a given path with arbitrary precision.
