I have visited various sites which claim a difference between stochastic kernels and conditional probability. However, I have read a paper which treats them the same, and the Wikipedia page on transition matrices actually lists a matrix full of conditional probabilities. The page linking to it claims that a Markov-kernel (stochastic-kernel, or probability-kernel) is simply an element of this transition matrix.
This is a contradiction in literature, and I would like some clarity on the issue. What is the difference between a stochastic kernel and a conditional probability statement?
It's possible they differ in generality alone, where the stochastic kernel is a specific case of conditional probability, but I haven't found any references on this.
The term conditional probability, is free of context, while stochastic kernel is used only when discussing stochastic processes.
A stochastic kernel is a specific type of probability density statement, while a conditional probability can be a statement which has nothing to do with stochastic processes.
The answer is curtsy of a comment by @Lee David Chung Lin.