Topics in Stochastic Analysis

PI将研究纯自然和应用自然随机分析中的几个问题。相互竞争的粒子系统出现在各种科学模型中，包括生物学和燃烧理论。PI将研究这些模型的各个方面，包括流体动力学极限，标记粒子行为和相关偏微分方程的性质。我们将研究由相同噪声驱动的反射布朗运动的长期行为。研究neumanneigenn函数的几何性质。证明了一些随机微分方程解的唯一性。将研究谐波函数的边界行为以及与跳跃过程相关的热核的局部和全局行为。研究分形集上的布朗运动。逆问题是应用科学中自然产生的数学模型，从地球物理学到医学和物理学。PI将研究与该模型相关的基础问题，例如相应方程解的存在性和唯一性。其他模型和过程将作为单独但相关研究的主题；它们包括以小时间尺度为条件的马尔可夫过程，具有4阶标度性质的过程的变量公式的变化和变阶的非局部算子。布朗运动和马尔可夫过程是各种自然现象的模型，如天气和气候、生物体的各种功能和金融市场。科学目标之一是根据现有数据做出准确的预测。必须发展数学方法，使这种预测成为可能和可靠。PI将研究导致一般结果的理论问题，以及可以作为一般方法测试平台的特定模型。理论研究的结果经常被用来在很好理解的模型中进行定量预测，并对复杂系统进行定量预测。PI的研究将主要集中在没有或很少显示长时间记忆的进程上。这种系统是无生命物质和人造系统的流行模型，但也适用于许多生物系统。

The PI's will investigate several problems in stochastic analysisof pure and applied nature. Competing particle systems appear invarious scientific models including biology and theory ofcombustion. The PI's will study various aspects of such models,including hydrodynamic limits, tagged particle behavior andproperties of the associated partial differential equations. Longterm behavior of reflected Brownian motions driven by the samenoise will be examined. Geometric properties of Neumanneigenfunctions will be investigated. Uniqueness of solutions tosome stochastic differential equations will be proved. Boundarybehavior of harmonic functions and local and global behavior ofheat kernel associated with jump processes will be examined.Brownian motion on a fractal set will be investigated. Inverseproblem is a mathematical model arising naturally in appliedsciences from geophysics to medicine and physics. The PI's willstudy the foundational questions related to this model, such asexistence and uniqueness of the solutions to the correspondingequations. Other models and processes will be the subject ofseparate but related studies; they include Markov processesconditioned on a small time scale, a change of variable formulafor processes with 4-th order scaling properties and non-localoperators of variable order.Brownian motion and Markov processes are models for a wide rangeof natural phenomena, such as weather and climate, variousfunctions of living organisms and financial markets. One of thescientific goals is to make accurate predictions based onavailable data. Mathematical methods have to be developed to makesuch predictions possible and reliable. The PI's will work on boththeoretical questions leading to general results and on specificmodels that can serve as testbeds for the general methods. Theresults of theoretical research are often used to makequantitative predictions in well undrstood models and to makequalitative predictions related to complex systems. The researchof PI's will be mostly focused on processes that display no orlittle long time memory. Such systems are popular models forinanimate matter and man-made systems, but also for manybiological systems.