Title page for ETD etd-06092009-162436


Type of Document Dissertation
Author Zito, Kevin W
Author's Email Address zito@math.lsu.edu
URN etd-06092009-162436
Title Convolution Semigroups
Degree Doctor of Philosophy (Ph.D.)
Department Mathematics
Advisory Committee
Advisor Name Title
Frank Neubrander Committee Chair
Charles Delzell Committee Member
Ricardo Estrada Committee Member
Richard Litherland Committee Member
Stephen Shipman Committee Member
Carter Hill Dean's Representative
Keywords
  • Laplace Transforms
  • Expected Value of Random Processes
  • Convolution Semigroups
  • Infinitely Divisible Distributions
Date of Defense 2009-05-06
Availability unrestricted
Abstract
In this dissertation we investigate, compute, and approximate convolution powers of functions (often probability densities) with compact support in the positive real numbers. Extending results of Ursula Westphal from 1974 concerning the characteristic function on the interval $[0,1]$, it is shown that positive, decreasing step functions with compact support can be embedded in a convolution semigroup in $L^1(0,infty)$ and that any decreasing, positive function $pin L^1(0,infty)$ can be embedded in a convolution semigroup of distributions. As an application to the study of evolution equations, we consider an evolutionary system that is described by a bounded, strongly continuous semigroup ${T(t)}_{tgeq0}$ in combination with a probability density function $pin L^1(0,infty)$ describing when an observation of the system is being made. Then the $n^{th}$ convolution power $p^{star n}$ of $p$ is the probability distribution describing when the $n^{th}$ observation of the system is being made and $E_n(x_0):=int_0^{infty}T(s)x_0,p^{star n}(s),ds$ is the expected state of the system at the $n^{th}$ observation. We discuss approximation procedures of $E_n(x_0)$ based on approximations of the semigroup $T$ (in terms of its generator $A$) and of $p^{star n}$ (in terms of its Laplace transform $widehat{p}).$

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