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

subsets of the natural numbers that does not converge a.e. even though the usual

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

submartingale could be decomposed into the sum of a right-continuous martingale and

an increasing process provided it was of "Class D," a uniform integrability condition

on the random variables obtained by stopping the submartingale at random times, but

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
and

D. Williams, *Diffusions, Markov Processes, and Martingales*,
J. Wiley, 1979.

- This problem was floating around the department in 1965. An obscure
problem,

known as Lauricella's problem, was solved in this paper. The problem involves finding

a polyharmonic function that satisfies conditions on the function and higher-order

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 15(1970)208-218.

- In 1957 and 1958, G.A. Hunt scooped the mathematical community by publishing

three papers in the Illinois Journal of Mathematics that made a quantum leap from

Brownian Motion and classical potential theory to the modern theory of Markov

processes and potential theory. In the second of these papers, he pointed out the lack

of a probability model for Markov processes incorporating creation of mass,

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 with creation

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 and several

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
of Microbiology

as a model for biochemical processes. A limiting process leads to a partial differential

equation involving a second-order elliptic differential operator and oblique derivative

boundary conditions.

Fennicae 17(1992)199-209.

8. Diffusion in a Polygon with Oblique Reflection, To be incorporated
in a book under

preparation.