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On a construction of Markov processes associated with time dependent Dirichlet spaces. (English) Zbl 0759.60081
Let $$(X,m)$$ be a locally compact, metric, separable space equipped with an everywhere densely positive Radon measure on $$X$$. Let $$(E,V)$$ be a regular Dirichlet form on $$L^ 2(X;dm)$$, then it is well-known by M. Fukushima [Trans. Am. Math. Soc. 162 (1971), 185-224 (1972; Zbl 0254.60055)] that there is a self-adjoint operator $$A$$ with domain $$D(A)\subset L^ 2(X)$$ such that $$E(u,v)=(-Au,v)$$ and such that the resolvent equation $(p-A)W_ pf=f,\quad\forall f\in L^ 2(X;dm), \tag{1}$ determines a Hunt process $$X(t)$$ on $$X$$. The author investigates a time dependent family of regular Dirichlet forms $$E^ \tau$$ $$(\tau\in R^ 1)$$ on $$L^ 2(X;dm)$$ with common domain $$V$$ and proves the existence of a Hunt resolvent $$(V_ p)$$ on the space-time state space $${\mathfrak X}=R^ 1\times X$$ such that, $\left(p- {\partial\over\partial\tau}-A^ \tau\right)V_ pf(\tau,x)=f(\tau,x),\quad \forall f\in L^ 2({\mathfrak X};dv),$ where $$dv=d\tau\cdot dm$$ and $$A^ \tau$$ is the generator associated with $$E^ \tau$$ via (1).
The main result of the paper is stated as follows: Theorem 4.2. There exists a Hunt process, $$M=(Y(t),P_{\tau,x})$$ on $${\mathfrak X}$$, such that $$E_{\tau,x}(\int^ \infty_ 0e^{-pt}f(Y(t))dt=V_ pf(\tau,x)$$ a.e. on $${\mathfrak X}$$. Moreover, if we decompose $$Y(t)$$ as $$Y(t)=(\tau(t),X(t))$$, then $$\tau(t)$$ is the uniform motion to the right on $$R^ 1$$.
The method used here is similar to that of symmetric Dirichlet form given by Fukushima and that of coercive non-symmetric Dirichlet form given by Ménandez, Lejan and Sylverstein. The author also uses the capacity method in order to prove the existence of quasi-continuous modification for elements in the domain of definition of the constructed time dependent form.
Reviewer: X.L.Nguyen (Hanoi)

##### MSC:
 60J45 Probabilistic potential theory 31C25 Dirichlet forms
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