Let $(X_t^x)_{t\in [0,\infty),\,x\in \mathbb{R}^n}$ be a Markov process taking values in $\mathbb{R}^m$ and defined on some stochastic basis $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\in [0,\infty}), \mathbb{P})$; where $X_0^x=x$ $\mathbb{P}$-a.s. Suppose also that $\sup_{t\geq 0} \mathbb{E}[\|X_t^x\|]<\infty$ for all $x \in \mathbb{R}^n$.

Let $f$ be the function sending $(t,x)\in [0,\infty)\times \mathbb{R}^n$ to the conditional law $\mathbb{P}(X_t^x)$. If $X_t^x$ has paths of finite $p$-variation a.s., for some fixed $p\geq 1$, then is $f$ Hölder-continuous? If so, how is the regularity of $f$ implies by the a.s. regulairty of its paths?

*Edit:* I quantify the distance between two laws by the Wasserstein distance on $\mathcal{P}_1(\mathbb{R}^m)$.

*Edit:* Under what (additional) conditions on $X_t^x$ do we have continuity?

set of conditional laws? $\endgroup$