Sensitivity Analysis of Nonlocal Operators withApplications to Jump Processe

许多现实世界的现象可以被认为是粒子在空间中的传输。这种传递可能是连续的，即弥漫性的，类似于花粉悬浮在介质中的不稳定运动（如布朗运动），也可能是不连续的，跳跃在时间上逐渐到达。后者发生，例如，当我们观察到粒子的俘获或隧穿。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，这些状态或观测值不能连续。有很好的论点认为，许多现实生活中的现象，如天气模式或股票价格，都表现出跳跃型行为。从数学的角度来看，我们可以使用l<s:1>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依赖于系统的状态。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，指定跳跃的强度不一定（唯一地）定义目标动态。在将其应用于理论和建模之前，我们需要付出相当大的努力来构建机制并描述其定性和定量性质。在这些应用中，我们关注于(i)内部数学应用，研究lsamvys型过程的定性性质(过程返回吗？它们移动到无穷远的速度有多快？我们能识别趋势吗？)，而且还涉及（ii）遍历性问题（在短时间内观察多个粒子是否与在长时间内观察一个粒子一样好？），这对实验科学至关重要，以及（iii）关于过程的统计和近似的应用问题。我们的建议的一个关键特点是，我们将使用先进的方法，从偏微分方程到lsamvy型过程的理论，反之亦然。因此，我们结合了弗罗茨瓦夫工业大学（Bogdan）和德累斯顿工业大学（Schilling）两个国际公认团队的专业知识，致力于概率分析和分析方法的互补领域。，跳跃过程和随机分析。

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.