Stability and Approximations of Symmetric Diffusion Semigroups and Kernels

Abstract

Let (Pt)t≥0and (Pt)t≥0be two diffusion semigroups on Rd(d≥2) associated with uniformly elliptic operatorsL=∇·(A∇) and L=∇·(A∇) with measurable coefficientsA=(aij) and A=(ãij), respectively. The corresponding diffusion kernels are denoted bypt(x,y) and pt(x,y). We derive a pointwise estimate on pt(x,y)- pt(x,y) as well as anLp-operator norm bound, wherep∈[1,∞], forPt-Ptin terms of the localL2-distance betweenaijandãij. This implies in particular that pt(x,y)- pt(x,y) converges to zero uniformly in (x,y)∈Rd×Rdand that theLp-operator norm ofPt-Ptconverges to zero uniformly inp∈[1,∈] whenaij-ãijgoes to zero in the localL2-norm for each 1≤i,j≤n. © 1998 Academic Press.