Detailed Balance: What is the continuous analogy of the transition matrix?

Question

I am having trouble understanding the definition of detailed balance in the case of a continuous state space. The definition of detailed balance that I am working from is:

A pmf $\pi$ on $\mathcal{X}$ satisfies detailed balance with respect to $T$ if $$ \pi_a T_{ab} = \pi_b T_{ba} \ \ \forall a,b \in \mathcal{X} $$

As I understand it, $T$ is the transition matrix and $T_{ab}$ represents the probability of moving from state $a$ to state $b$. I am struggling with the notion of $T$ in the case where $\mathcal{X}$ is continuous. Say for example that $\mathcal{X} = \mathbb{R}$, then wouldn't $T_{ab}=0$ as a single value in $\mathbb{R}$ has measure 0? Also, what is the structure of the transition matrix $T$? Does it have an uncountably infinite number of rows and columns? How is this possible?

Answer 1

Detailed balance for a continuous Markov chain with transition kernel $K(\cdot,\cdot)$ and stationary density $f(\cdot)$ writes as $$f(x)K(x,y)=f(y)K(y,x)$$ It means that, in a stationary regime, the joint distribution of $(x_t,x_{t+1})$ is the same as the joint distribution of $(x_{t+1},x_{t})$. This implies that the chain $(x_t)$ is (time-)reversible.