Induced Dirichlet forms and capacitary inequalities
Let (X,\(\Sigma\),m) be a locally compact space equipped with a positive Radon measure m with support X. Let \(\xi\) (.,.) be a Dirichlet form on \[ (1)\quad\Gamma(\xi)\times\Gamma(\xi)\subset L^ 2(X;m)\times L^ 2(X;m), \] and define \[ \exp_\theta (A):=\begin{cases} \inf\{\xi_\theta(m,m):u\in\Gamma_ A\} & \text{if \(\Gamma_ A\neq\varnothing\)} \\ +\infty & \text{if \(\Gamma_ A=\varnothing\)} \end{cases} \] where \(A\subset X\) is open and \[ \xi_{\theta}(u,v)=\xi (u,v)+\theta (u,v),\quad u,v\in \Gamma (\xi), \] \[ \Gamma_ A=\{u\in \Gamma (\xi):\;u\geq 1\quad m.a.e.\text{ on } A\}. \] cap\(_{\theta}(.)\) is called the \(\theta\)-capacity. Now let \(F:X\to X^*\) be a proper map of X onto a (usually smaller) locally compact, separable, Hausdorff space \(X^*\) and let \(m^*\) be the image of m under F (which is supposed to have support \(X^*).\) We may construct a form \(\xi^*\) on \(\Gamma (\xi^*)\times \Gamma (\xi^*)\subset L^ 2(X^*,m^*)\times L^ 2(X^*,m^*)\) by defining \[ (2)\quad \xi^*_ 0(u,v)=\xi (u.F,v.F),\quad u,v\in \Gamma (\xi^*_ 0), \] where \(\Gamma (\xi^*_ 0)=\{u\in L^ 2(X^*,m^*)| u.F\in \Gamma (\xi)\}\). In the first section the authors study conditions on the expectation operator \({\mathcal F}_ m=E_ m(.| F)\) s.t. \(\xi^*\) is a Dirichlet form. The notion of core map F is introduced in order to include the assumption ``no killing''. In the case where \(A=F^{-1}(A^*)\), \(A^*\subset X^*\) open, the following estimate is obtained: \[ (3)\quad {\mathcal P}_ m(\tau \leq 1)\leq e^{\theta}.E_ m(e^{-\theta_{\tau}})\leq \theta^{- 1}.e^{\theta}.cap_{\theta}(A)\leq \theta^{- 1}.e^{\theta}.cap^*_{\theta}(A^*)\leq \] \[ \theta^{- 1}.e^{\theta}.\min [\xi^*_ 0(u,u)| u\in \Gamma_ A]\leq \theta^{-1}.e^{\theta}.\xi^*_ 0(v,v), \] for any suitable choice of \(v\in \Gamma_ A\). In the case \(X^*\subset R\) the original problem is transferred to the one for a one-dimensional process. In particular when \(A^*=(a,\infty)\) we are addressing the problem of estimating, \[ P_ m(\sup_{t\in [0,T]}F(x(t))>a), \] where \(F:X\to R\) is some given function and X(.) is a general conservative m-symmetric diffusion. In the two last sections 2 and 3 the authors estimate the right-hand side of (3) for a given reversible diffusion x(.) in \(R^ n\) and for \(f=|.|\). The following asymptotic limit is obtained: \[ \lim_{\iota \to \infty}\nu (\iota).P_ m(\tau^*\leq T)=T. \] Thus a result of \textit{G. F. Newell} [see J. Math. Mech. 11, 481-496 (1962; Zbl 0115.136)] is generalized and interpreted. The authors used mainly the techniques of stochastic calculus for diffusions for the computation of the above mentioned results.
scientific article; zbMATH DE number 4169765