Selected Publications and Comments
1. Mean Convergence of Martingales, Trans. Amer. Math. Soc. 87(1958)439-436.
In 1950, J. Dieudonne constructed an example of a martingale on the
filter of finite
2. Class D Supermartingales (with Guy Johnson, Jr.), Bull. Amer. Math.
subsets of the natural numbers that does not converge a.e. even though
boundedness condition on norms holds. This left open the possibility
of convergence in
mean for martingales on filters or directed sets.
In 1963, Paul A. Meyer proved that a continuous parameter right-continuous
3. Biharmonic Functions and Brownian Motion, J. Appl. Probability
submartingale could be decomposed into the sum of a right-continuous
an increasing process provided it was of "Class D," a uniform integrability
on the random variables obtained by stopping the submartingale at random
left open the question of whether or not each uniformly integrable
submartingale is of
Class D. An example of a uniformly integrable submartingale that is
not of Class D is
given in this paper.
"The Helms-Johnson example. This a first look at one of
the most celebrated
counterexamples in the subject, one to which we return..."--L.C.G.Rogers
D. Williams, Diffusions, Markov Processes, and Martingales,
J. Wiley, 1979.
This problem was floating around the department in 1965. An obscure
4. Markov Processes with Creation of Mass, Z. Wahr. und Verw. Gebeite 7(1967)225-234.
known as Lauricella's problem, was solved in this paper. The problem
a polyharmonic function that satisfies conditions on the function and
Laplacians of the function at the boundary of an arbitrary domain.
The problem was
solved using probabilistic methods.
5. Markov Processes with Creation of Mass, II, Z. Wahr. und Verw. Gebeite
In 1957 and 1958, G.A. Hunt scooped the mathematical community by publishing
6. Stochastic Formation of Hierarchies, J. Appl. Probability 10(1973)27-38.
three papers in the Illinois Journal of Mathematics that made a quantum
Brownian Motion and classical potential theory to the modern theory
processes and potential theory. In the second of these papers, he pointed
out the lack
of a probability model for Markov processes incorporating creation
annihilation of mass having been incorporated into his work. This lack
was overcome in
4 above by constructing an infinite measure describing a Markov process
of mass. Only conditional probabilities are allowed in this context.
Further studies of
these processes were carried out by Talma Leviatan in her 1970 thesis
subsequent papers. The construction of such processes was reported
by S. Kuznetsov
several years later (Teoria Veroyatin. i ee Primen. 18(1973)596-601;
Engl. Trans. in
Theory Prob. Appl. 18(1974)).
This work was first suggested by Prof. Carl Woese of the Department
7. Brownian Motion in a Closed Convex Polygon with Normal Reflection, Ann.
as a model for biochemical processes. A limiting process leads to a
equation involving a second-order elliptic differential operator and
8. Diffusion in a Polygon with Oblique Reflection, To be incorporated
in a book under