Potential theory related to some multiparameter processes
In the first part a potential theory is constructed from a continuous kernel on a locally compact metrizable space. This includes a study of capacity, quasi-continuity, equilibrium measures and potentials. In the second part the author presents the example of a kernel \(U\) on \(\mathbb{R}^d\), \(d\geq 1\), which is defined using \(n\) convolution semigroups \((\pi^j_t)_{t \geq 0}\), \(1 \leq j \leq n\), of symmetric probability measures on \(\mathbb{R}^d\) \[ Uf = f* v_1* \cdots* v_n \quad \text{where} \quad v_j = (1/ \sqrt \pi) \int^\infty_0 t^{-1/2} e^{-t} \pi^j_t dt. \] In the third and main part the probabilistic counterpart is discussed in detail. It is a multiparameter process \((Z_t^x)_{t \in \mathbb{R}^n_+}\) where \(Z^x_{(t_1, \dots, t_n)} = x + X^1_{t_1} + \cdots + X^n_{t_n}\) and \(X^1, \dots, X^n\) are \(n\) independent processes given by \((\pi^1_t)_{t \geq 0}, \dots, (\pi^n_t)_{t \geq 0}\), respectively.
scientific article; zbMATH DE number 763780
scientific article