  We propose periodic Macdonald processes as a $(q,t)$-deformation of periodic
Schur processes and a periodic analogue of Macdonald processes. It is known
that, in the theory of stochastic processes related to a family of symmetric
functions, the Cauchy-like identity gives an explicit expression of a partition
function. We compute the partition functions of periodic Macdonald processes
relying on the free field realization of the Macdonald theory. We also study
several families of observables for periodic Macdonald processes and give
formulas of their moments. The technical tool is the free field realization of
operators that are diagonalized by the Macdonald symmetric functions, where the
operators admit expressions by means of vertex operators. We show that, when we
adopt Plancherel specializations, the corresponding periodic Macdonald process
is related to a Young diagram-valued periodic continuous process. It is known
that, for periodic Schur processes, determinantal formulas are only available
when we extend them to take into account the charge. We also consider this kind
of shift-mixed periodic Macdonald processes and discuss their Schur-limit.
