Multidimensional Stochastic Analysis

基础上，首席调查员将研究两个概率领域的问题。第一个问题与哈纳克不平等有关。Harnack不等式断言偏微分方程解的非负性在点满足某些有界性估计，从而允许从全局信息获得逐点估计。它们在偏微分方程中被用来获得热核估计和证明解的正则性。在概率论中，它们被用来获得某些随机过程的转移密度估计和正则性。主要研究人员将调查何时可以获得与非局部算子有关的函数的Harnack不等式。所讨论的运算符有一个积分项，并对应于带有跳跃的过程。第二个研究领域涉及数学生物学中的种群模型产生的随机微分方程解的唯一性。这些方程描述了随着粒子数量的增加，每个粒子的质量减少，分支速率增加，分支扩散过程的极限。允许一个粒子的分枝速率和扩散机制取决于系统中的所有其他粒子。分枝扩散被用作多种物种的种群动力学模型。得到的方程通常要么是无限维的，要么退化的，要么两者兼而有之。首席研究员将继续他的工作，证明这些方程的唯一性。长期以来，物理和生物科学中的许多系统都可以用随机过程来建模。最近，人们发现，许多金融和经济系统也可以这样建模。为了研究更复杂的系统，出现了新类型的随机过程。举个例子，股票价格通常被视为依赖于一个连续的随机过程，即布朗运动。然而，由于战争、新发现等原因，股票价格的波动往往会有突然的跳跃。因此，也有必要研究具有跳跃的随机过程。在研究人口模型时，人们预计人口的行为将根据人口是大是小而有质的不同。主要研究者的研究主要涉及两种类型的随机过程，一种是带有跳跃的随机过程，如股票市场的例子；另一种是关于可能退化的系统的随机过程，如人口例子。正在研究的一些问题是，该方程是否只有一个解，以及该解是否具有足够的正则性，以便为模型提供新的信息。

0244737Bass The principal investigator will be working on problems in two areas of probability. The first is concerned with Harnack inequalities. A Harnack inequality asserts that nonnegative solutions to a partial differential equation satisfy certain boundedness estimates at points, and thus allow one to obtain pointwise estimates from global information. They are used in partial differential equations to obtain estimates on heat kernels and to prove regularity properties of solutions. They are used in probability to obtain transition density estimates and regularity properties of certain stochastic processes. The principal investigator will investigate when one can obtain Harnack inequalities for functions related to non-local operators. The operators in question have an integral term and correspond to processes with jumps. The second area of research concerns uniqueness for the solutions of stochastic differential equations arising from population models in mathematical biology. These equations describe the limit of branching diffusion processes as the number of particles increases, the mass of each particle decreases, and the branching rate increases. The branching rate and the diffusion mechanism for a particle are allowed to depend on all other particles in the system. Branching diffusions are used as models of population dynamics for a large variety of species. The equations that result are typically either infinite dimensional, degenerate, or both. The principal investigator will continue his work on proving uniqueness for these equations. It has been known for a long time that many systems in the physical and biological sciences can be modeled by stochastic processes. More recently it has been discovered that many financial and economic systems can also be so modeled. To investigate more complex systems, new types of random processes have arisen. To give an example, stock prices are often viewed as depending on a continuous random process, Brownian motion. Yet the fluctuations of stock prices often have sudden jumps, resulting from wars, new discoveries, etc. Thus it is essential to also study stochastic processes with jumps. When studying population models, one expects that the behavior of the population will be qualitatively different depending on whether the population is large or whether it is small. The research of the principal investigator is primarily concerned with two types of stochastic processes, ones with jumps, as in the stock market example, and ones concerning systems that can degenerate, as in the population example. Some of the questions that are being investigated are whether there is only one solution to the equation and whether the solution has sufficient regularity to be useful in providing new information for the model.