# Weaker than the Markov property but better than nothing?

Given a probability space, we say that $$(X_t)_{t \geq 0}$$ is Markov w.r.t its own filtration $$(\mathcal F_t)$$ if for all $$s,

$$P(X_t \in \cdot | \mathcal F_s) = P(X_t \in \cdot | X_s)$$

Are you aware of weaker but similar properties such as

$$P(X_t \in \cdot | \mathcal F_s) = P(X_t \in \cdot | X_0, X_s)$$

or more interestingly,

$$P(X_t \in \cdot | \mathcal F_s) = P(X_t \in \cdot |X_0, X_s, \int_0^s X_u du )$$

If so, what are the names of such properties and where could I read about them? If it is not present in the literature, can you give an example of a non-trivial process satisfying

$$P(X_t \in \cdot | \mathcal F_s) = P(X_t \in \cdot |X_0, X_s, \int_0^s X_u du )$$