I have a question about Hunt processes and its equivalence.


I'm reading *Dirichlet Forms and Symmetric Markov Processes* by M. Fukushima, Y. Oshima, and M. Takeda. The following theorem is stated in this book.


In the following, $X$ be a locally compact separable metric measure space and $m$ a Radon measure on $E$ with $\text{supp}[m]=E$.



> 
> [Theorem 4.2.4] Let $\mathbb{M}\_1$ and $\mathbb{M}\_2$ be two
>  $m$-symmetric Hunt processes on $X$ possess a common regular Dirichlet
>  form on $L^{2}(X;m)$. Then, $\mathbb{M}\_1$ and $\mathbb{M}\_2$ are
>  equivalent. That is, there is a common properly exceptional set $N$
>  such that 
>  \begin{align\*} p\_{t}^{1}f(x)=p\_{t}^{2}f(x),\quad x \in X
> \setminus N,\,t>0, 
> \end{align\*} 
>  where $f$ is a bounded Borel measurable
>  function on $X$, $p\_{t}^{i}$ is the transition function of
>  $\mathbb{M}\_{i}$, $i=1,2$.
> 
> 
> 


**Question**


Under what conditions does the following hold?


For any bounded Borel function $f$ and $t>0$, 
\begin{align\*} 
(\ast)\quad p\_{t}^{1}f(x)=p\_{t}^{2}f(x),\quad x \in X ,\,t>0.
\end{align\*}


There is a sufficient condition for $(\ast)$: 



> 
> [Proposition 4.1 in [this book](https://link.springer.com/chapter/10.1007%2FBFb0074090)]
>  Suppose that both $\mathbb{M}\_1$ and $\mathbb{M}\_2$ satisfy the absolute
>  continuity condition: for any $t>0$ and $x \in X$, 
>  \begin{align\*}
> p\_{t}^{1}(x,\cdot)\text{ and }p\_{t}^{2}(x,\cdot)\text { are absolutely
> continuous w.r.t. }m.
> \end{align\*} Then, $(\ast)$ holds.
> 
> 
> 


The author says that this claim follows immediately from [Theorem 4.2.4], but I did not understand it... If you know, please let me know.



Hunt processes and its equivalence
I wonder what Fukushima really meant, but the statement – as written – appears to be false: think of a diffiuson on $\mathbb{R} \setminus \{0\}$ for which $0$ is a non-exit entrance point on both sides, and extend this process to $\mathbb{R}$ in two ways, letting the process go either to the right or to the left when it starts at $0$.


To be specific, take the $2$-dimensional Bessel process: let $Y\_t$ and $Z\_t$ be independent standard Brownian motions, and define $X\_t^{(1)} = ((x + Y\_t)^2 + Z\_t^2)^{1/2}$ if the starting point $X\_0^{(1)} = x$ is positive and $X\_t^{(1)} = -((x + Y\_t)^2 + Z\_t^2)^{1/2}$ if $x \leqslant 0$. Define $X\_t^{(2)}$ in the same way, but change the conditions $x > 0$ and $x \leqslant 0$ to $x \geqslant 0$ and $x < 0$, respectively. Then $X\_t^{(1)}$ and $X\_t^{(2)}$ are equivalent in the sense of Dirichlet forms, satisfy the absolute continuity condition, but their laws are different when started at $0$.


