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?



Regularity of law of conditional law of a Markov process equivalent to regularity of its paths
No, this has no reason to be true. Take for example $X\_t^x = x+t$ for $x \ge 0$ and $x-t$ for $x < 0$ ($n=1$). Paths are smooth, but $f$ is discontinuous at $x=0$.


