  We study transitions between the Floquet states of a periodically driven
oscillator caused by the coupling of the oscillator to a thermal reservoir. The
analysis refers to the oscillator that is driven close to triple its
eigenfrequency and displays resonant period tripling. The interstate
transitions result in a random ``walk'' over the states. We find the transition
rates and show that the walk is nonlocal in the state space: the stationary
distribution over the states is formed by the transitions between remote
states. This is to be contrasted with systems in thermal equilibrium, where the
distribution is usually formed by transitions between nearby states. The
analysis of period tripling allows us to explore the features of the
multi-state Floquet dynamics including those missing in the previously explored
models of driven oscillators such as the absence of detailed balance for low
temperatures. We use the results to study switching between the period-3 states
of the oscillator due to quantum fluctuations and find the scaling of the
switching rates with the parameters.
