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 AbstractIn 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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