Stochastic Processes and Their Applications

1)在此，我们假设，冰川循环可以完全解释为大气，存储和埋藏状态之间的碳转移。这一理论建立在冰川埋藏假说和温室气体效应的基础上。我们通过开发一个封闭的随机模型来支持我们的假设，该模型可以在统计上重现过去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：相对陡峭的上升冰川状态相比，衰退到，目前延长全新世间冰期，和有点神秘的变化，从古老的41 ka世界较浅，更高频率的冰川循环到目前的100ka深循环世界。2)我们将获得超过程（这里是测度值马尔可夫过程）的表示，这对分析和模拟都是有益的。超过程在过滤理论、分支过程和种群遗传学中是一种高密度、高活动极限的粒子过程。然而，该限制失去了：a）粒子表示和B）联合分布信息（在不同时间上）。库尔茨和他的合作者（Ocone，Donnelly，Xiong，Rodrigues）引入了能级和马尔可夫映射，允许粒子及其谱系在极限中的无限集合。然而，这种方法仍然比统一理论更聪明。另外，Billingsley研究存在的概率措施对应的联合分布，但没有人解决的问题：当一个超过程的投影随机措施的路径空间？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度。3)脊分解和表示方法并不是强大数定律（SLLN）和大偏差原理（LDP）的灵丹妙药，因为许多超过程分别缺乏紧支撑性质和随机方程表示。在45年来，渡边的经典SLLN分支马尔可夫过程只有孤立的SLLN超过程在欧几里德空间被发现。我们将从时间和粒子近似的角度研究超过程的SLLN、CLT、LDPs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.