# Burgess Davis

Purdue University

## Talk 1:

Title: Random walk and brownian motion peturbed at extrema

Let b(t) be brownian motion. Define r(t) by
r(0)=a>0, dr(t)=db(t) except when r(t)=0,, where r reflects,
-and when r(t)=max{r(s):s-1 is a constant. This process looks like a reflected
brownian motion with an upwards or downwards kick at maxima,
depending on whether q<0 or q>0, a very naive model for
a pure buy(or sell)-high stock market. It also arises as
a limit of self-interacting random walks. I will discuss
this process, especially questions about its stability,
and related processes.

## Talk 2:

Title: Distribution of brownian local time on curves

It is well known that the distribution of
brownain local time at zero, up to time one, has the
distribution of the the absolute value of a standard normal
variable. I will show that the distribution of the local
time on the graph of a smooth increasing function on
[0,1] is distributionally smaller than twice this local
time at zero, and that this "twice" may not be replaced with
a smaller multiple.