Introduction and Main Results

This book provides a careful and accessible exposition of the functional analytic approach to the problem of construction of strong Markov processes with Ventcel’ (Wentzell) boundary conditions in probability theory (see the last Chap. 26 ). More precisely, we prove existence theorems for (Feller semigroup) Feller semigroups with Dirichlet boundary condition, oblique derivative boundary condition and first order Ventcel’ boundary condition for second order, uniformly elliptic differential operators with discontinuous coefficients. Our approach here is distinguished by the extensive use of the ideas and techniques characteristic of the recent developments in the Calderón–Zygmund theory of singular integral operators with non-smooth kernels (see [33–35, 140, 166, 167]). It should be emphasized that there is an approximate functional calculus for such operators resembling the Fourier transform apparatus (see [22–26, 28, 137]).