This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5).
The behavior of harmonic functions and potential functions for processes that have jumps will be studied. In particular, how does smoothness in the coefficients of the operators translate to the smoothness of harmonic and potential functions? Secondly, uniqueness for several probability models that have origins in mathematical physics and mathematical biology will be investigated. One model is that of stochastic partial differential equations in one dimension with a space-time white noise as a driving term. A second model is that of superprocesses with branching interactions.
In recent years researchers in mathematical physics, economics, and mathematical finance have realized that to adequately model real-world phenomena, the possibility of jumps must be allowed. For example, an unexpected discovery or unexpected regional conflict might cause a sudden jump up or down in stock prices. However, some of the very basic questions about models that incorporate jumps are as yet unanswered. One of the areas to be investigated is the regularity of such models. A typical question is: if the initial data is perturbed slightly, is it true that the behavior of the model at future times will also be only slightly perturbed?
