  Borrowing ideas from open quantum systems, we describe a formalism to encode
ensembles of trajectories of classical stochastic dynamics in terms of
continuous matrix product states (cMPSs). We show how to define in this
approach "biased" or "conditioned" ensembles where the probability of
trajectories is biased from that of the natural dynamics by some condition on
trajectory observables. In particular, we show that the generalised Doob
transform which maps a conditioned process to an equivalent "auxiliary" or
"driven" process (one where the same conditioned set of trajectories is
generated by a proper stochastic dynamics) is just a gauge transformation of
the corresponding cMPS. We also discuss how within this framework one can
easily prove properties of the dynamics such as trajectory ensemble equivalence
and fluctuation theorems.
