In my research, I need to deal with a notion that can be called `dependency of channels of two random variables'. Informally, the dependency of the channels of $X$ and $Y$ means that $X (Y)$ becomes $X^\prime (Y^\prime)$ without being affected by $Y (X)$.
I would like to know the name of this notion, hopefully, with some references (books, papers or web sites).
Let me describe the detail of the dependency of channels with bit formal way.
We know that independency between two random variables is basic notion in stochastic theory. For independent random variables $X,Y$, we can decompose the distribution into the product of each distribution: $$P(X,Y) = P(X)P(Y),$$ where $P$'s are distributions of random variable(s).
A communication channel can be interpreted as a positive affine map of a distribution. Say, a cannel of $X,Y$ maps $P(X,Y)$ to $Q(X^\prime,Y^\prime)$ as follows: $$Q(X^\prime,Y^\prime) = \iint P(X^\prime,Y^\prime\mid X,Y)P(X,Y) dXdY $$
By combining those two notions, I think we can introduce a notion like `dependence of the channel of (X,Y)' as stated below.
Let us think the situation where we can rewrite the conditional distribution in the second equation as the product of two conditional distribution of $X$ and $Y$: $$P(X^\prime,Y^\prime\mid X,Y) = P(X^\prime\mid X)P(Y^\prime\mid Y)$$ Then, the channel can be thought of a model of the process converting $X (Y)$ to $X^\prime (Y^\prime)$ without being affected by $Y (X)$, i.e. only depending on $X (Y)$.
I think that this kind of channel can be called dependent channel or product channel etc. However, I could not find similar concept by searching web. Please help!