Dynkin's formula and large coupling convergence
Let \(H\geq 1\) be a selfadjoint operator in the Hilbert space \(\mathcal H\), let \(J\) be a bounded linear operator from \((D(\sqrt H),\|\sqrt H\bullet\|)\) into some auxiliary Hilbert space and denote for every \(\beta>0\) by \(H_\beta^J\) the nonnegative selfadjoint operator in \(\mathcal H\) defined via \[ \Bigl\|\sqrt{H_\beta^J} f\Bigr\|^2 = \|\sqrt H f\|^2 + \beta \|Jf\|^2, \qquad f\in D(\sqrt H). \] It is shown that the limit \(D_\infty^J = \lim_\beta D_\beta^J\), \(D_\beta^J := H^{-1} - (H_\beta^J)^{-1}\) exists strongly and satisfies the following generalized Dynkin formula: \[ D_\infty^J = P H^{-1} \] where \(P\) is the orthogonal projection in \(D(\sqrt H)\) onto \((\text{ker} J)^\bot\). Both the operator and trace class norms of \(D_\infty^J - D_\beta^J\) are explicitly given so that the exact rate of convergence can be studied. In the special case where \(H-1\) is the selfadjoint generator of the \(m\)-symmetric Dirichlet form \(\mathcal E\) on \(L^2(E,m)\) and where \(J\) maps \(D(\mathcal E)\) onto \(L^2(E,\mu_\Gamma)\), \(\mu_\Gamma\) being the equilibrium measure of the closed subset \(\Gamma\subset E\) having finite Dirichlet capacity, the estimate becomes \[ \|D_\infty^J - D_\beta^J\| \leq 1/(1+\beta),\quad\beta > 0; \] it is also shown that this estimate is sharp.
scientific article; zbMATH DE number 2141616