An analytic approach to the Fleming-Viot processes with interactive selection
Let \(E\) be the set of all probability measures on a Polish space \(S\) and \(m\) be a probability measure on the Borel \(\sigma\)-algebra on \(E\). Consider the form \[ {\mathcal E}^b (u,v) = \int_E \{\langle \nabla u(\mu), \nabla v(\mu)\rangle_\mu + \langle b(\mu), \nabla u(\mu)\rangle_\mu v(\mu) + \alpha u(\mu) v(\mu)\} m(d\mu) \] defined for \(u, v \in {\mathcal F}C^\infty_b\), where \(\nabla u(\mu) = (\nabla_x u(\mu))_{x\in S}\), \(\nabla_x u(\mu) = {d\over ds} u(\mu + s\varepsilon_x)|_{s=0}\), \(b(\mu)(x) = b(x,\mu)\) and \(\langle f, g\rangle_\mu = \int fgd\mu - \int fd\mu \int gd\mu\). In the first part of this paper, under the conditions that \(({\mathcal E}, {\mathcal F}C^\infty_b)\) is closable on \(L^2(E;m)\) and \(\text{sup}_{\mu\in E} |b(\mu) |_\mu < \infty\), the closability of \(({\mathcal E}^b_\alpha, {\mathcal F}C^\infty_b)\) is proved for some \(\alpha > 0\). Further, the closure becomes local and quasi-regular semi-Dirichlet form and hence an associated measure-valued diffusion exists. This result is applied for the Dirichlet form \(\mathcal E\) on \(L^2 (m^{Fv})\) determined by Fleming-Viot process with neutral mutation to get the new diffusion describing Fleming-Viot process with neutral mutation and interactive selection. Such process is also studied by \textit{T. Shiga} [J. Math. Kyoto Univ. 30, No. 2, 245-279 (1990; Zbl 0751.60044)]. More detailed analysis of the process associated with \({\mathcal E}^b\) is given. In particular, if \(m\)-a.e. \(\mu\) is purely atomic, then \(X_t\) is purely atomic and continuous in the variation norm. In the Fleming-Viot case, a necessary and sufficient condition for the exceptionality of the set \(\{\mu = \sum^k_{i =1} p_i \varepsilon_{x_i};\) \(x_i \in S\), \(p_i \geq 0\), \(\sum^k_{i= 1} p_i = 1\}\) is given in terms of the mutation measure \(\nu_0\) and intensity \(\theta\).
scientific article; zbMATH DE number 796530