Appl. Math. Optim. 9:177-191 (1982) Applied Mathematics and Optimization Integro-differential Operators Associated with Diffusion Processes with Jumps Suzanne Lenhart Department of Mathematics, University of Tennessee, Knoxville, TN 37996 Communicated by J. L. Lions Abstract. We show existence and W12~ p (3 Wl,°%regularity results for the integro-differential equation, associated with a diffusion process with jumps on a bounded domain. The second order elliptic partial differential operator and the integral operator involved here are both maximum principle type operators, which enables us to make W ~'°° a priori estimates. 1. Introduction We investigate here an integro-differential equation that arises from diffusion processes with jumps. We show the existence and WI~P A W 1,°°-regularity of the solution of this integro-differential equation, LU(X)-- fR.[U(X + z)-u(x)-vu(x).zXl,l¢,]m(dz ) = f(x) a.e. in fl, u = 0 outside ~, (1.1) where f~ is a bounded, smooth domain in R n, p > n >/2, Xlz141 is the characteristic function of the set (Izl ~< 1), m(dz) is a positive measure on R n such that fl~ Izl2m(dz)+flz Izlm(dz) < oo, (1.2) i~<1 I>l and Lu is a second order uniformly elliptic operator of the form Lu = - aijUxixj + biuxi .4- cu (summation convention). (1.3) 0095-4616/82/0009-0177 $03.00 ©1982 Springer-Verlag New York Inc.