  We first study probabilistic dynamical systems from logical perspective. To
this purpose, we introduce the finitary dynamic probability logic}
($\mathsf{DPL}$), as well as its infinitary extension
$\mathsf{DPL}_{\omega_1}\!$. Both these logics extend the (modal) probability
logic ($\mathsf{PL}$) by adding a temporal-like operator $\bigcirc$ (denoted as
dynamic operator) which describes the dynamic part of the system. We
subsequently provide Hilbert-style axiomatizations for both $\mathsf{DPL}$ and
$\mathsf{DPL}_{\omega_1}\!$. We show that while the proposed axiomatization for
$\mathsf{DPL}$ is strongly complete, the axiomatization for the infinitary
counterpart supplies strong completeness for each countable fragment
$\mathbb{A}$ of $\mathsf{DPL}_{\omega_1}\!$. Secondly, our research focuses on
the (frame) definability of important properties of probabilistic dynamical
systems such as measure-preserving, ergodicity and mixing within $\mathsf{DPL}$
and $\mathsf{DPL}_{\omega_1}$. Furthermore, we consider the infinitary
probability logic $\mathsf{InPL}_{\omega_1}$ (probability logic with initial
probability distribution) by disregarding the dynamic operator. This logic
studies {\em Markov processes with initial distribution}, i.e. mathematical
structures of the form $\langle \Omega, \mathcal{A}, T, \pi\rangle$ where
$\langle \Omega, \mathcal{A}\rangle$ is a measurable space, $T: \Omega\times
\mathcal{A}\to [0, 1]$ is a Markov kernel and $\pi: \mathcal{A}\to [0, 1]$ is a
$\sigma$-additive probability measure. We prove that many natural stochastic
properties of Markov processes such as stationary, invariance, irreducibility
and recurrence are $\mathsf{InPL}_{\omega_1}$-definable.
