# Is there a regular Dirichlet form with no associated Feller process?

I'm reading Dirichlet Forms and Symmetric Markov Processes by M. Fukushima, Y. Oshima, and M. Takeda (hereafter, [FOT]). In Chapter 7, where they discuss the construction of a Markov process associated to a Dirichlet form, there is the statement:

"In general, it is hopeless to construct a Feller transition function from the $L^2$-semigroup $T_t$ associated with the given Dirichlet space."

Is there an example of a regular symmetric Dirichlet form which is not associated to any Feller process?

and conversely

Are there useful conditions on a Dirichlet form which guarantee the existence of an associated Feller process?

Here is some further background.

Let $X$ be a locally compact Hausdorff space (or even a separable metric space) equipped with a Radon measure $\mu$. Suppose $X_t$ is a Markov process on $X$, with transition function $p(t,x,\Gamma) = \mathbb{P}_x(X_t \in \Gamma)$. For a measurable function $f$ on $X$, write $P_t f(x) = \int_X f(y) p(t, x, dy) = \mathbb{E}_x[f(X_t)]$.

Suppose also that $\mathcal{E}$ is a symmetric Dirichlet form on $L^2(X,\mu)$, with domain $\mathcal{F}$, whose corresponding $L^2$ strongly continuous contraction semigroup is $T_t$. $\mathcal{E}$ and $X_t$ are associated if $T_t f = P_t f$, $\mu$-a.e., for all $f \in L^2(X,\mu)$.

It is a well-known theorem (see [FOT]) that if $(\mathcal{E}, \mathcal{F})$ is regular (i.e. $\mathcal{F} \cap C_c(X)$ is $\mathcal{E}_1$-dense in $\mathcal{F}$ and uniformly dense in $C_c(X)$), then there exists a Markov process $X_t$ associated to $\mathcal{E}$. Moreover, $X_t$ is a Hunt process: it is normal ($\mathbb{P}_x(X_0 = x) = 1$), strong Markov, and quasi-left continuous (essentially, it is càdlàg and the jumps are unpredictable).

Now $X_t$ is said to be Feller if, for any $f \in C_0(X)$, we have $P_t f \in C_0(X)$. (Here $C_0(X)$ is the continuous functions on $X$ which vanish at infinity; i.e. the uniform closure of $C_c(X)$.) I am asking when it is possible, given a regular Dirichlet form $\mathcal{E}$, to choose an associated Hunt process which is in fact Feller.

The tricky part is that, given $\mathcal{E}$, the associated Hunt process $X_t$ is not unique, so even if one can find an associated process which is not Feller, there could be another one which is Feller.

For instance, let $X = \mathbb{R}^n$, $n \ge 3$, with $\mu = m$ Lebesgue measure, and let $\mathcal{E}(f,g) = \frac{1}{2}\sum_{i=1}^n \partial_i f \partial_i g$ be the classical Dirichlet form. If $B_t$ is standard Brownian motion, then $B_t$ is clearly associated to $\mathcal{E}$. Let $X_t$ also be a Brownian motion, but with the origin $0$ changed to an absorbing state. With $n \ge 3$, $\{0\}$ is a polar set, so if $X_t$ starts at any point $x \ne 0$, then almost surely it never hits $0$, so $\mathbb{P}_x(X_t \in \Gamma) = \mathbb{P}_x(B_t \in \Gamma)$ for $x \ne 0$. Thus it is clear that $X_t$ is also associated to $\mathcal{E}$. $X_t$ is not Feller, but $B_t$ is, so this is not the sort of counterexample I am looking for.

Thanks for any information!

Actually, it looks like a slight modification of my example works. Let $X = \mathbb{R}^n \backslash \{0\}$, $\mu = m$ Lebesgue measure, $\mathcal{E}$ the classical Dirichlet form with its domain $H^1_0(\mathbb{R}^n \backslash \{0\})$ (i.e. Dirichlet boundary conditions). It's clear that $T_t f(x) = \int_{\mathbb{R}^n} f(y) \frac{1}{(2\pi t)^{n/2}} e^{-|x-y|^2/2t}dy$ (a.e.) which is not a.e. equal to a $C_0$ function (it doesn't vanish near $x=0$). So no Feller process can be associated with this Dirichlet form.