Let $\Omega$ a finite metric space with $\forall x,y,z\in \Omega:d(x,y)<d(x,z)+d(z,y)$. Does there exists a continuous Markov process $X$ on $\Omega$ such that $$\mathbb{E}_x(T_y)=d(x,y)$$ for all $x,y\in \Omega$ with $T_y:=\inf \{t\geq 0, X_t=y \}$ and $\mathbb{E}_x$ is for the conditon $X_0=x$?

Remark: since one can first go to $z$ and then to $y$ we have by the Markov property $\mathbb{E}_x(T_y)\leq \mathbb{E}_x(T_z)+\mathbb{E}_z(T_y)$ so the triangular inequality. But is it the only constraint?

So far, I tried to express $\mathbb{E}_x(T_y)$ using the resolvant of the stochastic matrix or to look for some connection with the resistance network (see for example the book "Markov Chain and Mixing Times" by Levin, Peres and Wilmer) but I haven't find anything very successful.

Let $\Omega$ a finite metric space with $\forall x,y,z\in \Omega:d(x,y)<d(x,z)+d(z,y)$. Does there exists a continuous-time Markov process $X$ on $\Omega$ such that $$\mathbb{E}_x(T_y)=d(x,y)$$ for all $x,y\in \Omega$ with $T_y:=\inf \{t\geq 0, X_t=y \}$ and $\mathbb{E}_x$ is for the conditon $X_0=x$?

Remark: since one can first go to $z$ and then to $y$ we have by the Markov property $\mathbb{E}_x(T_y)\leq \mathbb{E}_x(T_z)+\mathbb{E}_z(T_y)$ so the triangular inequality. But is it the only constraint?

So far, I tried to express $\mathbb{E}_x(T_y)$ using the resolvant of the stochastic matrix or to look for some connection with the resistance network (see for example the book "Markov Chain and Mixing Times" by Levin, Peres and Wilmer) but I haven't find anything very successful.