Lipschitz-type inequalities for Markov kernels

Question

Let $K(\cdot\mid\cdot)$ be a Markov kernel. For a measure probability $\mu$, denote by $\mu K$ the probability measure induced by $K$ and $\mu$, i.e. $(\mu K)(A):=\int_\Omega \mu(d\omega)K(A\mid\omega)$ for each (Borel) measurable $A$.

For an appropriate notion of "distance" $d$ between distributions, one would like a result saying that that whenever $d(\mu,\nu)\leq \varepsilon$ then $d(\mu K,\nu K)\leq \varepsilon\cdot C$, for some constant $C$ not depending on $\varepsilon$.

The distance $d$ should satisfy some concentration bound implying, loosely speaking, that the distance between $\mu$ and the empirical measure obtained by $N$ i.i.d. samplings from $\mu$ approaches 0, as $N\rightarrow+\infty$.

I am aware of a result along these lines for the $\ell_1$-distance attributed to Dobrushin, see this question; but did not find a reference for that. Any (introductory) references to results concernig other types of distances would be much appreciated as well.

Answer 1

Assuming that your distance is convex, the question reduces just to the case when both $\mu$ and $\nu$ are delta measures, and amounts then to the inequality $$\tag {$\star$} d(\delta_x,\delta_y) \le C d(\kappa_x,\kappa_y) \;, $$ where $\kappa_x=\delta_x K$ denotes the transition probability of the kernel $K$ from a point $x$. If $d$ is the total variation distance (I would refrain from calling it $\ell^1$ distance), and $C=1$, then this is the aforementioned inequality attributed to Dobrushin. Condition ($\star$) is also quite natural (and has been considered) when $d$ is the transportation distance determined by a certain metric on the underlying space.