Internship Opportunity with Prof. Bruce Bayly, Math Dept., U. Arizona
Date Posted: Sep 1998

Description:

Differential equations arise in a great many applications of
mathematics to the physical world.  In a typical application, the system
of interest is modeled as a perfectly isolated from the rest of the
universe, obeying exactly-specified equations relating rates of change
to state variables.  If the initial conditions are also prescribed
exactly, then the entire future of the system is determined.  This type
of description is therefore called `deterministic'.  If the system is
stable (in an appropriate sense) then small perturbations from the outside
world will not invalidate the basic usefulness of a deterministic model.

If a system experiences large random perturbations from the outside world
(like a skyscraper in a hurricane or earthquake), or is intrinsically
unstable or chaotic (like a tumbling space structure, or a fluid flow,
or a national economy) then a deterministic description is far from
adequate.  The effect of perturbations may nevertheless be incorporated
reasonably well by simply adding a `white-noise' term to a deterministic
differential equation description, resulting in what is called a `stochastic
differential equation'.  Stochastic differential equations first arose in
economics (Bachelier) and physics (Einstein, Langevin, Fokker&Planck
at the beginning of the 20th century, and were put on a firm mathematical
footing by Kolmogorov and Ito by the 1950's.  They are now used in a huge
variety of models and applications throughout science and technology.

Since the `solution' of a stochastic differential equation is a random
quantity, in principle only the distribution of solutions can be predicted.
In practice even predicting the distribution is extremely difficult,
as it is governed by a partial differential equation in many variables.
There are many situations (e.g. fluid turbulence) in which even a
very coarse approximation to the distribution would be of great value.
At present Prof. Bruce Bayly (also Profs. Greg Eyink and David Levermore)
is exploring basic strategies for modeling distribution functions
of low-dimensional stochastic differential equations, and there is
plenty of scope for involvement by an interested undergraduat.

Requirements:

The student should have had an introduction to differential equations
and also computer programming.  A student who is a declared Math major
is entitled to use the Math Dept.'s computer facilities.
No prior knowledge of probability or
statistics is needed, although s/he should be prepared to learn some basic
material pretty quickly.  Willingness to explore ideas than may not pan out
is essential, and the ability to generate (and follow up) one's own ideas i
definitely desirable.

Outcome:
Final outcome is determined with the interested students on a case by case
basis. Although not required, can lead to a senior thesis project.