J. Evol. Equ. 11 (2011), 771–792 © 2011 Springer Basel AG 1424-3199/11/040771-22, published online May 25, 2011 DOI 10.1007/s00028-011-0110-6 Journal of Evolution Equations Multiplicative perturbations of the Laplacian and related approximation problems Francesco Altomare, Sabina Milella, and Graziana Musceo Abstract. Of concern are multiplicative perturbations of the Laplacian acting on weighted spaces of con- tinuous functions on R N , N ≥ 1. 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 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 applica- tions of such approximation formula are finally shown concerning both the semigroups and the associated Markov processes. 1. Introduction Multiplicative perturbations of the Laplacian of the form α play an important role in the theory of wave propagation in nonhomogeneous media whose density is related to the coefficient α. The parabolic problem associated with α and coupled with suitable initial- boundary conditions has been investigated (often via the theory of C 0 -semigroups of operators) in the setting of L 2 spaces or in spaces of continuous bounded functions defined on R N , N ≥ 1, or on a bounded domain of it with a smooth boundary. With- out any claim of completeness, we refer to this respect, e.g., to [2], [4, Section 6.3.9 and the references quoted in the Notes of Section 6.3, p. 477], [10, 11, 13, 17, 19]. In this paper, we study such perturbations in weighted spaces of (unbounded) contin- uous functions on R N . We show, indeed, that the operator α defined on its maximal domain is the pre-generator of positive quasicontractive C 0 -semigroup that fulfills the Feller property (i.e., it leaves invariant the space C 0 (R N ) and it is a contractive semigroup on it) and that is associated with a suitable probability transition function on R N and hence with a Markov process on R N . Mathematics Subject Classification (2010): 47D06, 47D07, 41A36 Keywords: Multiplicative perturbation, Laplacian, Positive semigroup, Weighted continuous space, Markov process, Approximation by positive operator, Integral operator. This work has been partially supported by the research project “Real Analysis and Functional Analytic Methods for Differential Problems and Approximation Problems”, University of Bari, 2010.