Analysis of multidimensional processes

考虑了两个概率领域的问题。第一部分涉及随机偏微分方程、具有分支和空间相互作用的测量值分支扩散、依赖于过去的方程和对应于退化椭圆算子的方程的区域中产生的随机微分方程解的唯一性。这些是典型的无限维方程组，通常具有在边界处退化的系数。第二个研究领域涉及哈纳克不等式和热核估计。我们将尝试在连续模型和跳跃模型中描述哈纳克不等式。此外，研究了有界区域上非局部算子方程解的正则性。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：如果初始数据受到轻微的扰动，那么模型在未来时间的行为是否也只会受到轻微的扰动？对于起源于数学生物学的某些模型，也存在类似的未解问题。在这些情况下，困难并不在于跳跃的存在，而是大量个体的存在以及个体之间可能的相互作用。

Problems in two areas of probability are considered. The first is concerned with uniqueness for the solutions of stochasticdifferential equations arising from the areas of stochastic partialdifferential equations, measure-valued branching diffusions with branchingand 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, 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 of the model at future times will also be only slightly perturbed? There are similar unanswered questions for certain models that originate in mathematicalbiology. In these cases the difficulty is not the presence of jumps, butthe presence of huge numbers of individuals and the possible interactionsbetween individuals.