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.