Sensitivity Analysis of Nonlocal Operators withApplications to Jump Processe

非局部算子的敏感性分析及其在跳跃过程中的应用

基本信息

项目摘要

Many real-world phenomena can be thought of as transportation of particles in space. The transport may be continuous, i.e. diffusive, similar to the erratic movement of pollen suspended in a medium (like Brownian motion) or discontinuous, with jumps gradually arriving in time. The latter occurs, e.g., when we observe trapping or tunneling of the particles. Another typical example are changes of a system with a finite number of states or observations, which cannot be continuous. There are good arguments that that many real-life phenomena, like weather patterns or stock prices exhibit jump-type behaviour. From a mathematical perspective, we may use Lévy-type processes which generalize continuous `diffusive’ phenomena since they allow for jumps and their behaviour depends locally on the state of the system. We want to capture fundamental features of jump dynamics and resolve essential difficulties by developing an abstract but flexible mathematical framework. This aims at offering a unifying perspective for complicated phenomena. It turns out that specifying the intensity of jumps does not necessarily (uniquely) define the target dynamic. We need to put considerable effort into the construction of the mechanism and describe its qualitative and quantitative properties before they can be applied in theory and modelling.Among the applications, we focus (i) on inner-mathematical applications studying qualitative properties of Lévy-type processes (Do the processes return? How quickly do they move to infinity? Can we identify trends?) but also on (ii) the question of ergodicity (is observing many particles for a short time as good as observing one particle for a long time?) which is paramount for experimental sciences, as well as (iii) applied questions on the statistics and approximation of the processes.A key feature of our proposal is that we will use advanced methods from partial differential equations to the theory of Lévy-type processes, and vice versa. Therefore, we combine the expertise of two internationally acknowledged teams, at TU Wroclaw (Bogdan) and TU Dresden (Schilling), working on the complementary fields of analysis and analytic methods in probability, resp., jump processes and stochastic analysis.
许多现实世界的现象可以被认为是粒子在空间中的传输。运输可以是连续的,即扩散的,类似于悬浮在介质中的花粉的不稳定运动(如布朗运动)或不连续的,跳跃逐渐到达时间。后者发生,例如,当我们观察到粒子的捕获或隧穿时。另一个典型的例子是具有有限数量的状态或观测值的系统的变化,这不可能是连续的。 有很好的理由认为,许多现实生活中的现象,如天气模式或股票价格表现出跳跃式的行为。从数学的角度来看,我们可以使用Lévy型过程,它概括了连续的“扩散”现象,因为它们允许跳跃,并且它们的行为局部地取决于系统的状态。我们希望捕捉跳跃动力学的基本特征,并通过开发一个抽象但灵活的数学框架来解决基本困难。这旨在为复杂的现象提供一个统一的视角。事实证明,指定跳跃的强度并不一定(唯一)定义目标动态。我们需要投入相当大的努力来构建的机制,并描述其定性和定量的性质之前,他们可以应用于理论和建模。在应用中,我们集中在(i)内部的数学应用研究定性性质的Lévy型过程(做的过程返回?它们移动到无穷远的速度有多快?我们能确定趋势吗?)而且还涉及(ii)遍历性问题(短时间内观察许多粒子与长时间观察一个粒子一样好吗?)这是最重要的实验科学,以及(iii)应用问题的统计和近似的过程。我们的建议的一个关键特点是,我们将使用先进的方法,从偏微分方程的理论Lévy型过程,反之亦然。因此,我们联合收割机结合了两个国际公认的团队的专业知识,在TU弗罗茨瓦夫(博格丹)和TU德累斯顿(席林),分别在概率分析和分析方法的互补领域工作,跳跃过程和随机分析。

项目成果

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Professor Dr. René Leander Schilling其他文献

Professor Dr. René Leander Schilling的其他文献

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{{ truncateString('Professor Dr. René Leander Schilling', 18)}}的其他基金

Transition density estimates for Lévy-type processes
Lévy 型过程的跃迁密度估计
  • 批准号:
    239237733
  • 财政年份:
    2013
  • 资助金额:
    --
  • 项目类别:
    Research Grants

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