Stability and Approximations of Symmetric Diffusion Semigroups and Kernels

✍ Scribed by Zhen-Qing Chen; Zhongmin Qian; Yaozhong Hu; Weian Zheng

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 483 KB
• Volume: 152
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.1997.3147

✦ Synopsis

Let (P t ) t 0 and (P t ) t 0 be two diffusion semigroups on R d (d 2) associated with uniformly elliptic operators L={ } (A{) and L ={ } (A {) with measurable coefficients A=(a ij ) and A =(a~i j ), respectively. The corresponding diffusion kernels are denoted by p 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 an L p -operator norm bound, where p # [1, ], for P t &P t in terms of the local L 2 -distance between a ij and a~i j . This implies in particular that | p t (x, y)& p~t(x, y)| converges to zero uniformly in (x, y) # R d _R d and that the L p -operator norm of P t &P t converges to zero uniformly in p # [1, ] when a ij &a~i j goes to zero in the local L 2 -norm for each 1 i, j n. 1998 Academic Press 1. INTRODUCTION Denote by R d (d 2) the d-dimensional Euclidean space. The minimal fundamental solution to the heat equation \ t &2 + u=0 on R d article no.