I am trying to understand the detailed balance condition precisely.
To get it I went on this page : https://cs.adelaide.edu.au/~paulc/teaching/montecarlo/node22.html
It is written the following :
Let $P(A,t)$ be the probability distribution to be in state $A$ at time $t$.
We have :
$$P(A,t+1)=P(A,t)+\sum_B W(B \rightarrow A)P(B,t) - W(A \rightarrow B)P(A,t)$$
Where $W(B \rightarrow A)$ is the probability that the system goes from state $B$ to state $A$.
I don't understand this formula.
I would write this :
$$P(A,t+1)=\sum_B P(B,t) W(B \rightarrow A,t)$$
Indeed, consider I am at time $t+1$. The only things I need to know the probability of where I can be at time $t$ are given by the previous states + the probability of transition.
I don't get why we would substract things like the formula above...?