Clock Systems in Stochastic System Theory

Classical (closed) stochastic systems

Definition.A closed stochastic system consists of a measurable space

X

along with a stochastic kernel

φ : X \to Δ (X)

, where

Δ (X)

is the measurable space of probability measures on

X

Classically, what we call the update map of a closed stochastic system is often called the law for a Markov process

Informal definition.A stochastic process satisfying the law

φ : X \to Δ (X)

consists of a sequence

x_{1}, x_{2}, \dots

X

-valued random variables such that

P (- ∣ x_{i} = x) = φ (x)

Properly elaborating the above definition involves some measure theory