# Regular Dirichlet form and the associated transition kernel

I am reading a paper by Fukushima "On a stochastic calculus related to Dirichlet forms and distorted Brownian motions" and support it by a book "Dirichlet forms and symmetric Markov processes" by Fukushima, Oshima and Takeda.

Let me also remind that a dense subset $\tilde{C}$ of $C_0(X)$ is said to be a core of a Dirichlet form $\mathcal{E}$ if $\tilde{C}$ is $\mathcal{E}_1$-dense in $D[\mathcal{E}]$. A Dirichlet form possessing a core is called regular.

In the paper, there are the following 2 sentences:

"Let $\mathcal{E}$ be a Dirichlet form and $T_t$ the associated semigroup of Markovian symmetric operators on $L^2(X;m)$. If $\mathcal{E}$ is regular, then $T_t$ can be realized as \begin{equation} T_t\,f(x) = \int_X P_t(x, dy) f(y) \end{equation} by a transition function $P_t(x, E)$ on $X$ which is $m$-symmetric in the sense that $\int_XP_t\,f(x)\,g(x)\,m(dx) = \int_X P_tg(x)\,f(x)\,m(dx)$."

I could not find the above result in the book and I do not understand why we need the regularity assumption for that. Namely, we know that the space $C_0(X)$ of continuous functions with compact support is dense in $L^2(X,m)$, so we by Riesz-Markov-Kakutani there exists a transition function $P_t(x,dy)$ such that the above equation holds for $f\in C_0(X)$, and by density we can extend it to $L^2(X;m)$. Please tell me why this is wrong. Thank you very much.

The problem with your argument, in two words, is null sets. $T_t$ is an operator on $L^2$, and an element of $L^2$ isn't a function, it's an equivalence class mod a.e. equality. You would like to say "For each $x \in X$, consider the linear functional on $C_0(X)$ defined by $f \mapsto T_t f(x)$, and then use Riesz-Markov-Kakutani to find the measure associated to this functional and call it $P_t(x, \cdot)$." But $T_t f$ is only given as an element of $L^2$, and so $T_t f(x)$ isn't well defined; it depends on which representative of $T_t f$ you choose.
Although the regularity imposed in Fukushima's theory is quite natural, it is a bit of overkill if all one wants is a transition semigroup $(P_t)_{t\ge 0}$. There is work culminating in a short paper of S.E. Kuznetsov (1987-Prob. Theory and its Application; http://epubs.siam.org/doi/pdf/10.1137/1131031) that can be applied to a general Dirichlet form, the only extra condition being needed is that the state space is "nice" -- standard Borel is enough. Picking an initial distribution $\mu$ that is equivalent to the symmetry measure $m$, one can use the $L^2$ operators $T_t$ to define the finite dimensional distributions of a stochastic process, and then invoke Kolmogorov's theorem to gain the existence of such a process, call it $X=(X_t)$ on some probability space $(\Omega,\mathcal F,\Bbb P)$ such that $\Bbb P[X_0\in B]=\mu(B)$, $$\Bbb P[X_s\in A,X_t\in B]=\int_X T_s(1_AT_{t-s}1_B)\,d\mu,\quad 0\le s<t,$$ etc. By the basic Dirichlet form theory, the semigroup $(T_t)$ is strongly continuous, so the process $X$ is necessarily stochastically continuous. This permits the application of Kuznetsov's result to obtain a (measurable in $(x,t)$) system $P_t(x,dy)$ of transition probabilities that satisfy the Chapman-Kolmogorov equation and represent $T_t$ as in the statement of your problem. The proof involves the Ray-Knight compactification, and builds on earlier work of J.B. Walsh dating to the early '70s.