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$\varepsilon:L^2(X,m) \to [0,\infty]$ is a strongly local, symmetric Dirichlet form generating a Markov semigroup $(P_t)_{t\ge0}$ in $L^2(X,m)$. Let $D(\varepsilon)=\{f\in L^2(X,m):\varepsilon(f)<\infty\}$.

We know that when $f\in L^2(X,m)$ , $P_tf \to f$ in $L^2(X,m)$. But when $f \in D(\varepsilon)$ ,why is $P_tf \in D(\varepsilon)$ and $P_tf \to f$ in $D(\varepsilon)$ ?

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There is another approach, maybe more intrinsic, using basically the quadraticity and the lower semicontinuity: we know that $\varepsilon (P_tf)$ is decreasing, in particular $\epsilon (P_t f) \leq \varepsilon (f)$. Furthermore by the quadraticity assumption we know also that $ \varepsilon ( P_tf + f ) \leq ( \sqrt{\varepsilon (P_t f)} + \sqrt{\varepsilon (f)} )^2 \leq 4 \varepsilon (P_t f)$.

Now from the lower semicontinuity of $\varepsilon$ we know that $ \liminf \varepsilon (P_t f) \geq \varepsilon (f)$; along with the inequality before we get $\lim_{t \to 0} \varepsilon (P_t f) = \varepsilon (f)$. In the same way one can show $\varepsilon(P_tf + f) \to \varepsilon(f)$ and now by the parallelogram law we get precisely $ \varepsilon (P_t f-f) \to 0$.

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A hint: by the spectral theorem, it suffices to consider the case $\varepsilon(f) = \int f^2 h\,dm$ where $h$ is a nonnegative measurable function. In this case, we have $P_t f = e^{-th} f$.

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  • $\begingroup$ :If we just consider the case $\varepsilon$ is $W^{1,2}$ function on $R^n$. Then in the PDE theory for heat equations, there is such a property? $\endgroup$
    – wang mu
    Commented Oct 4, 2013 at 10:11
  • $\begingroup$ @wangmu: If I understand your question, yes. If we take $\varepsilon$ to be the classical Dirichlet form $\varepsilon(f) = \int |\nabla f|^2$ on $\mathbb{R}^n$, with domain $W^{1,2}$, so that $P_t$ is the heat semigroup, then for $f \in W^{1,2}$, we have $P_t f \to f$ in $W^{1,2}$-norm as $t \to 0$. In this case the application of the spectral theorem is just the Fourier transform so we can do everything explicitly. $\endgroup$
    – Nate Eldredge
    Commented Oct 4, 2013 at 15:26

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