Multiplicative perturbations of the Laplacian and related approximation problems

Of concern are multiplicative perturbations of the Laplacian acting on weighted spaces of continuous functions on \mathbbR^N,  N 3 1{{\mathbb{R}}^{N}, N\geq1} . It is proved that such differential operators, defined on their maximal domains, are pre-generators of positive quasicontractive C ₀-semigroups of operators that fulfill the Feller property. Accordingly, these semigroups are associated with a suitable probability transition function and hence with a Markov process on \mathbbR^N{{\mathbb{R}}^{N}} . An approximation formula for these semigroups is also stated in terms of iterates of integral operators that generalize the classical Gauss-Weierstrass operators. Some applications of such approximation formula are finally shown concerning both the semigroups and the associated Markov processes.