I am confronted with the definitions of a Lipschitz-Markov martingale and Lipschitz kernel. They are given as follows:
A kernel $k:x \mapsto \theta_x$ transporting $\mu$ to $\nu=\mu k$ is called Lipschitz if there exists a set $A\subseteq \mathbb{R}^d$ with $\mu(A)=1$ such that $k\vert_A$ is Lipschitz of constant 1 from $(A,\vert\vert\cdot\vert\vert_{\mathbb{R}^d})$ to $(\mathcal{P}(\mathbb{R}^d),W)$, where $\mathcal{P}(\mathbb{R}^d)$ is the set of probabilities on $\mathbb{R}^d$ and $W$ is the Kantorovich distance between two probabilities.
- Question: $\theta_x$ should be the disintegration wrt to what?
A process $(X_t)_t$ is a Lipschitz-Markov martingale if it is a Markovian martingale and the Markovian transitions are Lipschitz kernels.
- Question: Can someone explain these definitions with an easy example?