Let $(E,\mathscr{E})$ be a measurable space. Two transition of probabilities 
$p, q\colon E\times\mathscr{E}\to [0,1]$ are said to be in duality relative to 
a probability measure $m$ if for every pair of non-negative functions $f,g$ we have 
$$
\int (Pf) \,g\, dm=\int f (Qg)\, dm 
$$
where $P$ and $Q$ are the transfer operator relative to $p$ and $q$ respectively
($Pf(x)=\int f(y)\, p(x,dy)$) .


For a fixed transition of probability $p$, I would like to know if there are general conditions that guarantee the existence of a transition of probability 
in duality with respect to $p$ ( relative to some $m$).


I have a second question. In the case $E=\{1,\dots,k\}$, a probability transition $p$ on $E$ is completely determined by the matrix $P=(p\_{ij})$ given by $p\_{ij}=p(i,\{j\})$. If $p=(p\_{1},\dots,p\_{k})$ is a positive probability vector ($pP=p$, $p\_{i}>0$), then the probability transition $q$, given by 
$q\_{ij}=\frac{p\_{j}}{p\_{i}}p\_{ji}$, is in duality with $p$ relative to 
$m=p\_{1}\delta\_{1}+\dots+ p\_{k}\delta\_{k}$. 


In this case, for every $i$, the probability $q\_{i}:=q(i,)$ is absolutely continuous 
with respect to $m$ and for every $i$ the Radon-Nikodym derivative satisfies
$$ 
\frac{dq\_{i}}{dm}(j)=\frac{q\_{ij}}{p\_{j}}\leq C
$$
for every $j$, where $C$ does not depend on $i$ neither on $j$. 


I would like to know if there are other examples with those properties. That is, I would like to know if there are examples in the case $E$ infinite ( any topological space for instance) of two probabilities $p$ and $q$ in duality relative to some stationary measure
$m$ such that the Radom-Nikodym derivative $
\frac{dq\_{x}}{dm}
$
is uniformly bounded for $m$-almost every $x\in E$,
that is, there is $C>0$ such that for $m$-almost every $x$ we have
$$
\frac{dq\_{x}}{dm}(y)\leq C, 
$$
for every $y$ (or $m$-almost every $y$). 



Transition of probability in duality and its properties
$\newcommand{\de}{\delta}
\newcommand{\De}{\Delta}
\newcommand{\ep}{\epsilon}
\newcommand{\ga}{\gamma}
\newcommand{\Ga}{\Gamma}
\newcommand{\la}{\lambda}
\newcommand{\Si}{\Sigma}
\newcommand{\thh}{\theta}
\newcommand{\R}{\mathbb{R}}
\newcommand{\E}{\operatorname{\mathsf E}} 
\newcommand{\PP}{\operatorname{\mathsf P}} 
\newcommand{\EE}{\mathcal E}$ 


A construction more general than your "discrete" example is as follows. Suppose that there is a measure $m$ on the sigma-algebra $\EE$ such that for each $x\in E$ the measure $p\_x:=p(x,\cdot)$ is absolutely continuous with respect to $m$, with some density $k\_x=\frac{dp\_x}{dm}$. Suppose also that the function 
\begin{equation}
 E\times E\ni(x,y)\mapsto k(x,y):=k\_x(y) 
\end{equation}
is $\EE\otimes\EE$-measurable. Then, by the Tonelli theorem, for all nonnegative $\EE$-measurable functions $f$ and $g$ on $E$ one has 
\begin{align\*}
 \int dm\,g\,(Pf)&=\int m(dx)g(x)\int p(x,dy)f(y) \\ 
 &=\int m(dx)g(x)\int m(dy)k(x,y)f(y) \\ 
 &=\int m(dy)f(y)\int m(dx)k(x,y)g(x) \\ 
 &=\int m(dy)f(y)\int q(y,dx)g(x) \\ 
 &=\int dm\,f\,(Qg), 
\end{align\*}
where $q(y,dx):=m(dx)k(x,y)$ (that is, $q(y,A)=\int\_A m(dx)k(x,y)$ for all $y\in E$ and $A\in\EE$), so that for $q\_y:=q(y,\cdot)$ one has $\frac{dq\_y}{dm}=k(\cdot,y)$. So, if $k(x,y)\le C$ for some positive real constant $C$ and all $(x,y)\in E\times E$, then $\frac{dq\_y}{dm}\le C$. 


