Stochastic Processes and Their Applications

1)在此，我们假设冰川周期可以完全用大气、储存和埋藏状态之间的碳转移来解释。这一理论建立在冰川埋葬假说和温室气体效应的基础上。我们开发了一个封闭的随机模型来支持我们的假设，该模型可以统计地再现过去200万年来观测到的冰芯和海洋沉积物样本。 然后，我们使用这个模型来解释：与衰退进入冰川状态相比，冰川状态相对陡峭地上升，当前延长的全新世间冰期，以及从较浅、较高频率的古代41ka冰川周期到目前100ka深层周期的有点神秘的变化。 2)我们将得到超过程(这里是测值马尔可夫过程)的表示，这对分析和模拟都是有益的。在过滤理论、分枝过程和种群遗传学中，超过程是粒子过程的高密度、高活性极限。然而，该限制丢失了：a)粒子表示和b)联合分布信息(在不同时间)。库尔茨和他的合作者(奥科内、唐纳利、熊、罗德里格斯)引入了级别和马尔可夫映射，允许无限收集粒子及其谱系。然而，这种方法仍然是比统一理论更聪明的想法。另外，比林斯利研究了对应于联合分布的概率度量的存在性，但没有人解决这个问题：什么时候超过程是随机度量在路径空间上的投影？我们将研究这两种表示，并确定何时没有单一的路径空间随机测量。 3)对于强大数定律(SLLN)和大偏差原理(LDP)，脊柱分解和表示方法不是灵丹妙药，因为许多超过程分别缺乏紧支撑性和随机方程表示。自从Watanabe关于分枝马尔可夫过程的经典SLLN以来的45年里，只发现了欧氏空间上超过程的孤立SLLN。我们将从时间和粒子近似的角度研究超过程的SLLN、CLT、LDP和LILS。

1) Herein, we postulate that the glacial cycles can be completely explained by carbon transfer between atmospheric, stored and buried states. This theory builds upon the glacial burial hypothesis as well as the greenhouse gas effect. We support our hypothesis by developing a closed stochastic model that can statistically reproduce the observed ice-core and ocean-sediment samples over the past two million years. We then use this model to explain: the relatively steeper rise out of glacial states as compared to recession into, the current extended Holocene interglacial epoch, and the somewhat mysterious change from the ancient 41ka world of shallower, higher frequency glacial cycles to the current 100ka world of deep cycles. 2) We will obtain representations for superprocesses (Measure-valued Markov processes here) beneficial to analysis and simulation alike. Superprocesses, in filtering theory, branching processes and population genetic, are a high-density, high-activity limit of particle processes. However, the limit looses: a) the particle representation and b) the joint distribution information (over different times). Kurtz and collaborators (Ocone, Donnelly, Xiong, Rodrigues) introduced levels and Markov mappings that allow an infinite collection of particles and their genealogy in the limit. However, this method is still more clever ideas than a unified theory. Separately, Billingsley studies the existence of probability measures corresponding to joint distributions but nobody has addressed the question: When is a superprocess the projection of a random measure on pathspace? We will investigate both representations and determine when there is no single pathspace random measure. 3) The spine decomposition and the representation approach are not panacea for strong laws of large numbers (SLLN) and large deviation principles (LDP) as many superprocesses lack the compact support property and stochastic equation representation respectively. In the 45 years since Watanabe's classical SLLN for branching Markov processes only isolated SLLN for superprocesses over Euclidean space been discovered. We will investigate SLLNs, CLTs, LDPs and LILs for superprocesses in terms of time and particle approximation.