Stability and Approximations of Symmetric Diffusion Semigroups and Kernels

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.