J. Evol. Equ. 15 (2015), 131–147 © 2014 Springer Basel 1424-3199/15/010131-17, published online October 14, 2014 DOI 10.1007/s00028-014-0253-3 Journal of Evolution Equations On the sectoriality of a class of degenerate elliptic operators arising in population genetics Angela A. Albanese and Elisabetta M. Mangino Abstract. We study the sectoriality of a class of degenerate second-order elliptic differential operators in the space of continuous functions on the canonical simplex of R d . This kind of operators arises from the theory of Fleming–Viot processes in population genetics. In this paper, we deal with the class of degenerate second-order differential operators L d u (x ) = d  i , j =1 x i (δ ij − x j )∂ 2 x i x j u (x ) + d  i =1 b i (x )∂ x i u (x ), x ∈ S d , (0.1) where S d is the canonical simplex of R d consisting of all x ∈[0, 1] d such that x i ≥ 0, for i = 0,..., d (here, x 0 := 1 − ∑ d i =1 x i ), b := (b i ) d i =1 ⊂ C ( S d ) and b(x ),ν(x )≥ 0, for every x ∈ ∂ S d , denoting by ν the unit inward normal at ∂ S d . The operator (0.1) appears in the theory of Fleming–Viot processes as a generator of a Markov C 0 -semigroup defined on C ( S d ). Fleming–Viot processes are measure- valued processes that can be viewed as diffusion approximations of empirical processes associated with some classes of discrete time Markov chains in population genetics. We refer to [16, 17, 20] for more details on the topic. The operator (0.1) has been largely studied using an analytic approach by several authors in different settings, see [1–3, 6, 9–11, 13–15, 23, 25, 26] and the references quoted therein. The interest comes from the fact that the equations describing the diffusion processes are of degenerate type, and hence, the classical techniques for the study of (parabolic) elliptic operators on smooth domains cannot be applied. Indeed, the operator (0.1) degenerates on the boundary ∂ S d of S d and the domain S d is not smooth as its boundary presents edges and corners. The study of such type of degenerate (parabolic) elliptic problems on C ([0, 1]) started in the fifties with the papers by Feller [18, 19]. In the Feller theory for the one-dimensional case, it is shown that the behaviour of the diffusion process on the boundary constitutes one of its main characteristics. So, the appropriate setting for studying the equations describing the diffusion process is the space of continuous functions on the simplex S d . Mathematic Subject Classification: Primary 35K65, 35B65, 47D07; Secondary 60J35 Keywords: Degenerate elliptic second-order operator, Simplex, Analyticity, Fleming–Viot operator, Space of continuous functions.