Let $(Y_n)_{n \in \mathbb{N}}$ be a Markov Chain with transition probability $$p(x, dy) \sim N(x, \epsilon)$$ Show that $Y$ is reversible w.r.t to the lebesgue measure . What I have done is just simply used Fubinis theorem to write
$$\int_A p(x, B) dx = \int_A \int_B \frac{1}{\sqrt{(2\pi\epsilon)}} \exp(\frac{-(y-x)^2}{2\epsilon})dy dx = \int_B \int_A \frac{1}{\sqrt{(2\pi\epsilon)}} \exp(\frac{-(x-y)^2}{2\epsilon})dx dy = \int_B p(y, A) dy $$
Now the only two things I used are fubini and that the transition probability has a PDF w.r.t. the lebesgue measure.
Did I oversee something or is every Markov Chain which has a PDF w.r.t to a measure $\mu$ also reversible w.r.t to that? Are there generalizations of this?
Your reversible example has the very special feature that the kernel $p(x,dy)$ has a density $\phi(x,y)$ that is symmetric in $x$ and $y$, i.e., $\phi(x,y)=\phi(y,x)$. This is generally not true and the process would not always be reversible.