#### Publication Date

Spring 1-6-1971

#### Abstract

Let {V(t,ω), t ≥ O, ω ε Ω} be a diffusion process on the real line with infinitesimal operator 1/2σ^{2}(⋅)D^{2} + m(⋅)D. Markov processes {V_{n}, n = 1,2,....} on the real line are constructed in such a way that the paths of V_{n} are step functions with jump size n^{-1/2} and

P_{O }[lim sup |V_{n}(s)-V(s)| = 0] =1

n∞ 0≤s≤t,

where P_{O }assigns probability one to paths starting at the origin at t = 0.

Let {T_{V}(t), t≥0, vε R} be a family of linear contraction operators on a Banach space B. Suppose T_{V}(t)f is continuous in v for all t≥0, fεB, and T_{V}(t)T_{W}(s) = T_{W}(s)T_{V}(t) for all t,s≥0, v,wε R. Let A_{V} be the infinitesimal operator of T_{V}.

Abstract continued in dissertation.

#### Degree Name

Mathematics

#### Level of Degree

Doctoral

#### Department Name

Mathematics & Statistics

#### First Committee Member (Chair)

L. H. Koopmans

#### Second Committee Member

Richard Griego

#### Third Committee Member

Rueben Hersh

#### Document Type

Dissertation

#### Recommended Citation

Quiring, Donald. "Random Evolutions on Diffusion Processes." (1971). https://digitalrepository.unm.edu/math_etds/160