Professor Michael Marcus
City College of New York

First Lecture: April 1, 3:00--4:00 pm, Skiles 269

A surprising relationship between Levy processes and stationary Gaussian processes

Brownian motion is both a stationary Gaussian process and a Levy process but besides this important example these two classes of stochastic processes appear to be fundamentally different. Recent work has disclosed startling relationships between sample path properties of Gaussian processes and higher order Gaussian chaos processes and local times and continuous additive functionals of Levy processes and more general strongly symmetric Markov processes. This has led to many new results about all these stochastic processes. These will be discussed in a lecture for a general mathematical audience.

Second Lecture: April 2, 3:00--4:00 pm, Skiles 269

Renormalized self-interaction local times and Wick power Gaussian chaos processes

Using an extension of an important isomorphism theorem of Dynkin one can analyze n-fold self-intersection local times of symmetric Levy processes in terms of 2n-th order Wick power Gaussian chaos processes. The work, which is joint with Professor Jay Rosen, is an extension of earlier joint work in which necessary and sufficient conditions are obtained for the continuity of a family of continuous additive functionals of certain Levy processes in terms of similar conditions for related families of second order Gaussian chaos processes.


About the speaker

Michael B. Marcus received his Bachelors degree from Princeton University in 1957 and his Ph.D. from the Massachusetts Institute of Technology in 1965. He has been a Professor at Northwestern University and Texas A&M University and a visiting professor at several European universities. He is currently a professor at the City College of New York and the CUNY Graduate Center. He is a Fellow of the Institute of Mathematical Statistics and was a Gugenheim Foundation Fellow in 1993. His main mathematical interests are in Gaussian processes, probability in Banach spaces, random Fourier series and the interplay between additive functionals of Markov processes and Gaussian chaos processes.