I'm studying markovianity of quantum process and the two article that i'm reading right now refers to two different definition of classical markovianity. The first one is the definition reported also here, and it is also what I studied in class. The second one is essentially this. Both the article and Wikipedia refer to the same book, but there is no the proof of equivalence of the two definition.
My question is: how can I prove that are equivalent? It seems reasonable to me that 1 implies 2, but I have no idea how to prove. Implication 2 $\implies$ 1, I am not even sure is true.
Firstly I will prove general fact about Markow Chains and then I'll go to your definitions for the case of homogenous one.
Let $(X_n)$ be a discrete time process. I'll use the following notation:
$\mu$ will be the distribution of $X_0$ that is $\mu(a):= \mu(\{a\}) = \mathbb P(X_0=a)$.
$A_n(a,b) = \mathbb P( X_n = b | X_{n-1}=a)$ if $\mathbb P(X_{n-1}=a) > 0$ (that is $A_n$ can be interpreted as a matrix with entries - probabilities of certain direction in $n'$th step).
Now our definitions are:
(1) $(X_n)$ is a Markow Chain if $\mathbb P(X_{n+1} = a_{n+1} | X_n = a_n , ... ,X_0 = a_0) = \mathbb P(X_{n+1} = a_{n+1} | X_n = a_n)$ for any $a_0,...,a_{n+1}$ such that it makes sense ( $\mathbb P(X_n = a_n ,... , X_0 = a_0) > 0 $)
(2) $(X_n)$ is a Markow Chain if $\mathbb P(X_0=a_0,...,X_n = a_n) = \mu(a_0)A_1(a_0,a_1)A_2(a_1,a_2)...A_n(a_{n-1},a_n)$ for any $a_0,...,a_n$ such that it makes sense.
Proof:
Direction (1) $\Rightarrow$ (2)
We're work with induction. Note that $\mathbb P(X_0 = a_0) = \mu(a_0)$ by definition of $\mu$, so the induction basis holds. Assume it holds for $n-1$. Then for $n$ we get:
$$ \mathbb P(X_0=a_0,...,X_n=a_n) = \mathbb P(X_{n} = a_{n} | X_{n-1}=a_{n-1},...,X_0=a_0) \mathbb P(X_{n-1} = a_{n-1},...,X_0=a_0)$$
where we used definition $\mathbb P(A \cap B) = \mathbb P(A | B) \mathbb P(B)$. Now for the first term on the right apply definition (1) and for the second term apply induction step, to get:
$$ \mathbb P(X_0=a_0,...,X_n=a_n) = \mathbb P(X_n =a_n | X_{n-1}=a_{n-1})\mathbb P(X_{n-1}=a_{n-1} | X_{n-2} = a_{n-2})... \mathbb P(X_0=a_0)$$ what can be rewritten as $$ \mathbb P(X_0=a_0,...,X_n=a_n) = \mu(a_0)A_1(a_0,a_1)...A_n(a_{n-1},a_n)$$
Direction (2) $\Rightarrow$ (1)
Simplier, just definition of conditional probability (conditioned on event with prob. greater than $0$)
$$ \mathbb P(X_n=a_n | X_{n-1}=a_{n-1},...,X_0=a_0) = \frac{\mathbb P(X_n=a_n,...,X_0=a_0)}{\mathbb P(X_{n-1}=a_{n-1},...,X_0=a_0)}$$ And now use (2) for numerator and denominator to get $$ \mathbb P(X_n=a_n| X_{n-1}=a_{n-1},...,X_0=a_0) = A_n(a_{n-1},a_n) = \mathbb P(X_n = a_n | X_{n-1}=a_{n-1})$$
Homogenous case:
Note that in this case, we just have $A_n(a,b)$ is the same for every $n$, so we can just define $p(b|a) = A_1(a,b) = \mathbb P(X_1 = b | X_0=a)$ and $p(a) = \mathbb P(X_0=a)$ (that is we don't need to care about time when we want to go to different states, only about the states from which and where we go to).
Definition $1$ may be unchanged (or $\mathbb P(X_n = a_n | X_{n-1}=a_{n-1},...,X_0=a_0) = \mathbb P(X_1 = a_n | X_0=a_{n-1})$
Definition $2$ changes to $\mathbb P(X_n=a_n,...,X_0=a_0) = p(a_0)p(a_1|a_0)...p(a_n|a_{n-1})$ which is exactly the notation used in your definition.