Problems in two areas of probability are considered. The first is concerned with uniqueness for the solutions of stochastic differential equations arising from the areas of stochastic partial differential equations, measure-valued branching diffusions with branching and spatial interactions, equations that depend on the past, and equations corresponding to degenerate elliptic operators. These are typically infinite-dimensional systems of equations, often with coefficients that degenerate at a boundary. The second area of research concerns Harnack inequalities and heat kernel estimates. An attempt will be made to characterize the Harnack inequality in both continuous models and ones with jumps. In addition regularity of solutions to equations associated with non-local operators in bounded regions will be investigated.
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? There are similar unanswered questions for certain models that originate in mathematical biology. In these cases the difficulty is not the presence of jumps, but the presence of huge numbers of individuals and the possible interactions between individuals.
