315 / 0022-0396/01 $35.00 Copyright © 2001 by Academic Press All rights of reproduction in any form reserved. Journal of Differential Equations 176, 315–355 (2001) doi:10.1006/jdeq.2000.3976, available online at http://www.idealibrary.comon On Feller Semigroups Generated by Elliptic Operators with Integro-differential Boundary Conditions E. I. Galakhov and A. L. Skubachevskiı ˇ Department of Differential Equations, Moscow State Aviation Institute, Volokolamskoe Shosse 4, 125871, Moscow, Russia 1 1 This work was partially supported by the Russian Foundation of Basic Researches (Grant 99-01-00028) and INTAS (Grant 97-30551). The first author also gratefully acknowledges the support of the University of Trieste, where his work was completed through a grant from the Consorzio Internazionale dell’Universita ` di Trieste. Received June 16, 1999; revised July 20, 2000 In the 1950s, W. Feller gave a complete characterization of the analytic structure of one-dimensional diffusion processes [1, 2]. He proved that if an ordinary differential operator A is an infinitesimal generator of some contractive nonnegative semigroup, then a domain D(A) consists of func- tions satisfying integro-differential boundary conditions. Later such semigroups were called Feller semigroups. Conversely, if D(A) consists of functions with these nonlocal conditions, then A is an infinitesimal generator of some Feller semigroup. An analogous problem for multidimensional diffusion processes in a domain Q … R n was studied by A. D. Ventsel’ [3]. He described a general form of nonlocal conditions, which determine a domain of elliptic operator being an infinitesimal generator for a Feller semigroup. Such nonlocal conditions contain the values of a function and its derivatives at each point x ¥ “Q and an integral of this function over Q ¯ with respect to a Borel measure m(x,dy). The existence of a Feller semigroup for an elliptic operator with that domain is much more complicated problem. The first steps in this direction were made by K. Sato and T. Ueno [4]. They obtained sufficient conditions for existence of a Feller semigroup in so- called ‘‘transversal’’ case. In this case, roughly speaking, the nonlocal terms have lower order with respect to boundary operators. It was proved that existence of a Feller semigroup is equivalent to solvability of some auxiliary problem on a boundary. Thus sufficient conditions for existence of a Feller semigroup had implicit form. However later the results of [4] allowed to obtain different explicit forms of sufficient conditions for transversal case