On a construction of Markov processes associated with time dependent Dirichlet spaces.

*(English)*Zbl 0759.60081Let \((X,m)\) 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 MĂ©nandez, 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.

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 MĂ©nandez, 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)