  Stochastic processes are commonly used models to describe dynamics of a wide
variety of nonequilibrium phenomena ranging from electrical transport to
biological motion. The transition matrix describing a stochastic process can be
regarded as a non-Hermitian Hamiltonian. Unlike general non-Hermitian systems,
the conservation of probability imposes additional constraints on the
transition matrix, which can induce unique topological phenomena. Here, we
reveal the role of topology in relaxation phenomena of classical stochastic
processes. Specifically, we define a winding number that is related to topology
of stochastic processes and show that it predicts the existence of a spectral
gap that characterizes the relaxation time. Then, we numerically confirm that
the winding number corresponds to the system-size dependence of the relaxation
time and the characteristic transient behavior. One can experimentally realize
such topological phenomena in magnetotactic bacteria and cell adhesions.
