Diffusions on ``simple'' configuration spaces (Q2702446)

In their previous work [Osaka J. Math. 37, No. 2, 273-314 (2000; Zbl 0968.58028)], the authors constructed a large class of diffusion processes on the multiple configuaration space \(\bar\Gamma_E\), i.e., on the particle space over \(E\) with possibly several particles at the same point. Here, conditions are given under which it is possible to restrict the diffusion process to the \`\` simple'' configuration space \(\Gamma_E\) that consists of all configurations in \(\bar\Gamma_E\) with at most one particle per point. As a first step towards this end, a condition is established which guarantees that a measure on \(\bar\Gamma_E\) does not charge the complement of \(\Gamma_E\). The second step consists in an examination of Dirichlet forms for which the complement of \(\Gamma_E\) is exceptional. By the general theory, the Markov process associated with such a Dirichlet form will a.s. never leave \(\Gamma_E\). The final section is devoted to the case study where the underlying space \(E\) is the free loop space over Euclidean space, i.e., where particles consist of random loops.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00048].