  We perform a systematic symmetry classification of the Markov generators of
classical stochastic processes. Our classification scheme is based on the
action of involutive symmetry transformations of a real Markov generator,
extending the Bernard-LeClair scheme to the arena of classical stochastic
processes and leading to a set of up to fifteen allowed symmetry classes. We
construct families of solutions of arbitrary matrix dimensions for five of
these classes with a simple physical interpretation of particles hopping on
multipartite graphs. In the remaining classes, such a simple construction is
prevented by the positivity of entries of the generator particular to classical
stochastic processes, which imposes a further requirement beyond the usual
symmetry classification constraints. We partially overcome this difficulty by
resorting to a stochastic optimization algorithm, finding specific examples of
generators of small matrix dimensions in six further classes, leaving the
existence of the final four allowed classes an open problem. Our symmetry-based
results unveil new possibilities in the dynamics of classical stochastic
processes: Kramers degeneracy of eigenvalue pairs, dihedral symmetry of the
spectra of Markov generators, and time reversal properties of stochastic
trajectories and correlation functions.
