Kalman filter and Markov chains?

Question

fuzzy logic

Hello!
After reading the publications on Habré on Kalman filters, including the Unscented Kalman filter, a question arose, it was partially raised in the discussions of one of the posts, but I would like to understand it in more detail.
The questions are:
1. If the process under study cannot be described by a linear function and it is stochastic, how can the Kalman filter be applied to it?
2. Is it possible to replace the matrix A from the extrapolation formula Xk = Ak*Xk-1 + e with the matrix of transition probabilities of Markov chains, so that, taking into account the dynamics of the process, predict the probability vector of states at the next step?
3. How can one move from the resulting predicted state probability vector to specific scalars? (Perhaps apply the apparatus of fuzzy logic? Are there ready-made analytical solutions?)
Probably, some of the answers are here , but this material is difficult to understand for me for various reasons.
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Thank you!