Absolute continuities of exit measures and total weighted occupation time measures for super-\(\alpha\)-stable processes (Q2495544)

Let \(D\) be a bounded smooth domain in \({\mathbb R}^d\), and \(\tau\) be the first exit time of the underlying symmetric \(\alpha\)-stable process \(\xi =\) \(\{ \xi_t, \Pi_x\), \(t \geq 0, x \in {\mathbb R}^d \}\) from the domain \(D\), where \(0 < \alpha < 2\). Let \(X =\) \(\{ X_t, X_{\tau}, Y_{\tau}, P_{\mu} \}\) be a super-\(\alpha\)-stable process in \({\mathbb R}^d\), where its branching rate is given by \(dt\), and its branching mechanism is of the form \(\Psi(z) = z^{1+ \beta}\) with \(0 < \beta \leqslant 1\). Here \(X_{\tau}\) (resp. \(Y_{\tau}\)) denotes the exit measure of \(X\) in \(D\) (resp. the total weighted occupation time measure of \(X\) in \(D\)). The purpose of this paper is to discuss the absolute continuity of \(X_{\tau}\) and \(Y_{\tau}\). Let \({\mathcal M}_c(E)\) denote the set of all finite measures on \({\mathcal B}(E)\) with compact support, and \({\mathcal M}_0(E)\) be the set of all finite measures on \({\mathcal B}(E)\) with finite points support. The authors prove that, for every \(\mu \in {\mathcal M}_c(D)\), \(X_{\tau}\) is \(P_{\mu}\)-a.s. absolutely continuous with respect to the Lebesgue measure \(dz\) on \(\bar{D}\), by making use of a convergence theorem on the fundamental solutions of the integral equation \[ u(x) + \Pi_x \int_0^{\tau} u^{1+ \beta} ( \xi_s) ds = H_D \nu(x), \quad x \in D, \] with \(\nu \in {\mathcal M}_0(\bar{D})\), where \(H_D\) is the Poisson operator of the process \(\xi\) in \(D\). Moreover, the \(P_{\mu}\)-a.s. absolute continuity of \(Y_{\tau}\) with respect to the Lebesgue measure \(dy\) on \(D\) is also derived with the help of a similar type of convergence theorem on the fundamental solutions of the integral equation \[ u(x) + \Pi_x \int_0^{\tau} u^{1+ \beta} ( \xi_s) ds = G_D \nu(x), \quad x \in D \setminus N_{\nu}, \] with \(N_{\nu} = \{ x ; \,\, G_D \nu(x) = \infty \}\) and \(\nu \in {\mathcal M}_0(D)\), where \(G_D\) is the Green operator of the process \(\xi\) in \(D\). The proofs are due to the standard analysis and the potential theory with the use of continuity of regularization, the strong Markov property and the Laplace transfrom. As to other closely related works, see \textit{D. A. Dawson} and \textit{K. Fleischmann} [J. Theor. Probab. 8, No.~1, 179--206 (1995; Zbl 0820.60032)] and \textit{Y. Ren} [Sci. China, Ser. A 43, No.~5, 449--457 (2000; Zbl 0962.60035)].