#### 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= R^{d} fx+h-fx -1fx˙h1 h≤1 ax,h h^{d+a} dh and consider the space-time process *Y _{t}* = (

*X*), where

_{t}, V_{t}*X*is the process that corresponds to the operator L , and

_{t}*V*=

_{t}*V*

_{0}+

*t.*Under the assumption that 0 <

*k*

_{ 1}≤

*a*(

*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*: Z

^{d}→R and consider the following operators defined by Lfx= y≠xfy -fx Ax,y x-y

^{d+a}, 3f,g x=x∈Z

^{ d}y≠ xfy -fx gy-gx Ax,y x-y

^{d+a}, and Gf,f x=y≠x fy-fx

^{2}Ax,y x-y

^{d+a}. ^ Under the assumption that 0 <

*k*

_{1}≤

*A*(

*x, y*) ≤

*k*

_{2}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 . ^

#### Recommended Citation

Huynh, Tho, "Parabolic Harnack inequality and Caccioppoli inequality for stable-like processes" (2009). *Doctoral Dissertations*. AAI3393013.

https://opencommons.uconn.edu/dissertations/AAI3393013