Title

Parabolic Harnack inequality and Caccioppoli inequality for stable-like processes

Date of Completion

January 2009

Keywords

Mathematics

Degree

Ph.D.

Abstract

In the first chapter of this dissertation, we introduce the parabolic Harnack inequality and the Caccioppoli inequality for stable-like processes. ^ In the second chapter, we let L be the operator defined by Lfx= Rd fx+h-fx -1fx˙h1 h≤1 ax,h hd+a dh and consider the space-time process Yt = (Xt, Vt), where Xt is the process that corresponds to the operator L , and Vt = V0 + t. Under the assumption that 0 < k 1a(x, h) ≤ k 2 and a(x, h) = a( x, –h), we prove a parabolic Harnack inequality for non-negative functions that are parabolic in a domain. We also prove some estimates on equicontinuity of resolvents. ^ In the third chapter, we let f : Zd→R and consider the following operators defined by Lfx= y≠xfy -fx Ax,y x-yd+a, 3f,g x=x∈Z dy≠ xfy -fx gy-gx Ax,y x-yd+a , and Gf,f x=y≠x fy-fx 2Ax,y x-yd+a . ^ Under the assumption that 0 < k1 A(x, y) ≤ k2 and A(x, y) = A(y, x), we establish a Caccioppoli inequality for powers of non-negative functions that are harmonic with respect to L . ^