By a Markov measure you mean, that for some stochastic kernel $K$ on the measurable space $(A,2^A)$ it holds that $P$ is the induced measure on $A^\Bbb N$?

What does an open set men in your context?

Do you mean $\langle f,\mu\rangle := \int_S f(x)\mu(\mathrm dx)$ for any $\mu \in M(S)$ and bounded Borel $f$?

Do you assume a fixed probability space (=dependence structure between $X$ and $Y$), or you only know distributions?

There is a way to solve this problem even for more general Markov processes, but could you please tell me more precise about your conditioning. In your case $\mathsf P(Y_k\geq 0\;\forall k) = 0$ so I guess you are interested rather in $$ \mathsf P(M_n \geq m|Y_k\geq 0 \; \forall 0\leq k\leq n) $$ am I right?

@Paul: I'm not sure whether it's true. Let $X\equiv \frac 12$ and let $p = 2$, then LHS is $\frac14$ and RHS is $\frac1{16}$, so your philosophy does not apply at least in such case.

@Paul: are you sure you have to take $p$ power in the RHS two times? Or you are just asking the question in the math.stackexchange version?

@Anthony: I agree - and I am pretty sure that one come up with a maximal coupling of $P$ and $\hat P$ doing it sequentially - I just wondered whether it's possible for the maximal coupling which satisfies one more additional assumption (aka $\gamma$-coupling according to Lindvall).

As far as I've checked your example, it is correct - thanks again. I guess, I shall accept you answer - but could you suggest how to compute $\Bbb P(X_1 = \hat X_1)$ or maybe you know how to express $\Bbb P(X\in A,\hat X\in \hat A)$?

Thank you. Do you have a reference for the proof of the last formula? It's like a Fubini theorem, but I've never seen a proof of its version for kernels. Also, in your first example, do you mean that $P$ and $\hat P$ in place of $\mu$ and $\hat \mu$ - just to keep it consistent with the notation in OP?