Parabolic Harnack inequality and Caccioppoli inequality for stable-like processes

Date of Completion

January 2009






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 . ^