Stability and Approximations of Symmetric Diffusion Semigroups and Kernels

Let (P_t)_t≥0and (P_t)_t≥0be two diffusion semigroups on R^d(d≥2) associated with uniformly elliptic operatorsL=∇·(A∇) and L=∇·(A∇) with measurable coefficientsA=(a_ij) and A=(ã_ij), respectively. The corresponding diffusion kernels are denoted byp_t(x,y) and p_t(x,y). We derive a pointwise estimate on p_t(x,y)- p_t(x,y) as well as anL^p-operator norm bound, wherep∈[1,∞], forP_t-P_tin terms of the localL²-distance betweena_ijandã_ij. This implies in particular that p_t(x,y)- p_t(x,y) converges to zero uniformly in (x,y)∈R^d×R^dand that theL^p-operator norm ofP_t-P_tconverges to zero uniformly inp∈[1,∈] whena_ij-ã_ijgoes to zero in the localL²-norm for each 1≤i,j≤n.