On the domain of a Fleming--Viot-type operator on an \(L^p\)-space with invariant measure (Q2902317)

One-dimensional Fleming-Viot processes are generated by second order differential operators \(L\varphi(x) = x(1-x)\varphi''(x)+(\alpha_1 (1-x) -\alpha_2 x)\varphi'(x)\) acting on \(C^2(0,1)\). These operators generate positive analytic \(C_0\)-operator semigroups \((T_t)\) on \(C[0,1]\), and \(C^2(0,1)\) is a core. The probability defined by \(d\mu_1(x)=\beta_1 x^{\alpha_1-1}(1-x)^{\alpha_2-1} d x\) is an invariant distribution (with norming constant \(\beta_1\)). \((T_t)\) is canonically extended to \(L^p([0,1],\mu_1)\), \(1\leq p<\infty\), as analytic \(C_0\)-semigroup. The authors are interested in the domain of the corresponding infinitesimal generator, which will be denoted by \(L_p\). This domain is characterized in Theorem 1, showing that a weighted Sobolev space is a core for \(L_p\).NEWLINENEWLINEFor \(N\in\mathbb{N}\), \(N\)-dimensional analogues of Fleming-Viot operators \(L_p\) are introduced as sums of (commuting) operators \(L_p^{(i)}\) acting on the \(i\)-th coordinate, \(1\leq i\leq N\). Applying a result of \textit{G. Dore} and \textit{A. Venni} [``On the closedness of the sum of two closed operators'', Math. Z. 196, 189--201 (1987; Zbl 0615.47002)], the authors obtain in Theorem 2 a description of the domain also in the case \(N>1\).