Multidimensional Stochastic Analysis

[244737] bass首席研究员将研究两个概率领域的问题。第一个是关于哈纳克不等式。一个哈纳克不等式断言一个偏微分方程的非负解在点上满足一定的有界估计，从而允许人们从全局信息中得到点估计。它们在偏微分方程中被用来估计热核和证明解的正则性。它们在概率学中用于获得某些随机过程的转移密度估计和正则性。主要研究者将研究何时可以得到与非局部算子相关的函数的哈纳克不等式。所讨论的算子有一个积分项，并且对应于有跳跃的过程。第二个研究领域涉及数学生物学中由种群模型引起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、每个粒子质量的减少和分支速率的增加，分支扩散过程的极限。允许一个粒子的分支速率和扩散机制依赖于系统中所有其他粒子。分支扩散被用作大量物种种群动态的模型。得到的方程通常要么是无限维的，要么是简并的，要么两者兼而有之。首席研究员将继续他的工作来证明这些方程的唯一性。人们早就知道，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，出现了新的随机过程类型。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突然跳升。因此，研究有跳跃的随机过程是必要的。在研究人口模型时，人们期望人口的行为会因人口大或小而有质的不同。首席研究员的研究主要涉及两种类型的随机过程，一种是具有跳跃的随机过程，如在股票市场的例子中，另一种是关于可以退化的系统，如在人口的例子中。正在研究的一些问题是，方程是否只有一个解，以及该解是否具有足够的规律性，以便为模型提供有用的新信息。

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.