Stochastic differential equations: potential theory and uniqueness

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。将研究具有跳变过程的调和函数和势函数的行为。特别是，算符系数的平滑性如何转化为调和函数和势函数的平滑性？其次，研究了起源于数学物理和数学生物学的几种概率模型的唯一性。一种模型是以时空白噪声为驱动项的一维随机偏微分方程模型。第二个模型是具有分支相互作用的超过程模型。近年来，数学物理学、经济学和数学金融学的研究人员已经意识到，要充分模拟现实世界的现象，必须允许跳跃的可能性。例如，一个意想不到的发现或意想不到的地区冲突可能会导致股票价格的突然上涨或下跌。然而，关于包含跳跃的模型的一些非常基本的问题尚未得到解答。要研究的领域之一是这些模型的规律性。一个典型的问题是：如果初始数据受到了轻微的扰动，那么模型在未来的行为是否也只会受到轻微的扰动？

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 forprocesses that have jumps will be studied. In particular, how does smoothness in the coefficients of the operators translate tothe smoothness of harmonic and potential functions?Secondly, uniqueness for several probability models that have origins inmathematical 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, andmathematical finance have realized that to adequately model real-worldphenomena, the possibility of jumps must be allowed. For example, anunexpected discovery or unexpected regional conflict might cause a sudden jumpup or down in stock prices. However, some of the very basic questionsabout models that incorporate jumps are as yet unanswered. One of the areasto 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 ofthe model at future times will also be only slightly perturbed?