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We use the method of the classical potential theory to construct the semigroup of operators that describe a Feller process on the line by pasting together two diffusion processes that satisfy a nonlocal Feller–Wentzell type condition for the pasting. 1. Main notation and setting of the problem Let Di = {x : (−1)x > 0}, i = 1, 2, be two domains on the real line, let ∂D = {0} be the joint boundary for these domains, and let Di = Di ∪ ∂D be the closure of Di. If D is a domain in R (in particular, D = R) and D is its closure, then Cb(D) denotes the Banach space of bounded continuous functions φ(x) on D equipped with the norm ‖φ‖ = supx∈D |φ(x)| and C0(D) denotes the space of bounded uniformly continuous functions on D. In what follows, if φ ∈ Cb(R), then the restriction of φ to Di is denoted by φi. Let C2(Di) be the subset of Cb(Di) containing all the functions φ such that φi, φ ′ i, φ ′′ i ∈ C0(Di), i = 1, 2. Assume that a differential operator Li is defined in the domain Di, i = 1, 2, and that it acts on functions φ of C2(Di), (1) Liφ(x) = 1 2 bi(x) dφi dx2 (x) + ai(x) dφi dx (x), i = 1, 2, where bi(x) and ai(x) are bounded continuous functions on Di and, moreover, bi(x) ≥ 0. We further assume that each operator Li generates a homogeneous diffusion process on Di. Denote by C2(R) the subset of Cb(R) consisting of the functions φ(x) such that φi ∈ C2(Di), i = 1, 2, and L1φ1(0) = L2φ2(0). Now we introduce the operator L acting on functions φ of C2(R) according to the rule (2) Lφ(x) = { L1φ(x) if x ∈ D1, L2φ(x) if x ∈ D2. Then we introduce the additional pasting condition at the point x = 0 that reduces the operator L (its closure, more precisely) to the infinitesimal operator of the Feller semigroup on C0(R). This condition is written as (3) L0φ(0) ≡ q1φ(0−)− q2φ(0+) + ∫ D1∪D2 (φ(0)− φ(y))μ(dy) = 0. 2010 Mathematics Subject Classification. Primary 60J60.

Semigroups of operators that describe a Feller process on the line, which is the result of pasting together two diffusion processes