Mathematical Sciences: Stochastic Processes

Marcus和Rosen教授将继续研究高斯过程和对称马尔可夫过程的局部时间之间的关系，并通过Dynkin同构定理证明这一点。这个定理的一个紧凑形式将被用来尝试得到局部时的Borell型不等式。类似于他们最近的工作，显示了对称马尔可夫过程和相关高斯过程的局部时间的样本路径性质的等价性，他们将研究由测度索引的对称马尔可夫过程的连续加性泛函与由测度索引的相关高斯混沌之间的关系。这需要对测量索引高斯过程和混沌进行研究。他们还将尝试扩展他们最近关于对称Levy过程和随机漫步的局部时间的迭代对数定律的工作，以涵盖二维整数格上循环随机漫步的有趣情况。Marcus教授和Rosen教授将继续研究高斯过程和对称马尔可夫过程的局部时间之间的关系。直到最近，这些随机过程才被认为是完全不相关的，除了在布朗运动的情况下，一种关系是由Dynkin最近的同构定理暗示的。Marcus教授和Rosen教授利用这种同构性来表明，这两类基本的随机过程之间存在着深刻的联系，它们都是无数连续时间随机现象的应用模型。这项研究可能会带来关于我们随机环境的重要新知识。

Professors Marcus and Rosen will continue their investigations of the relationship between Gaussian processes and the local times of symmetric Markov processes put into evidence by the Dynkin Isomorphism Theorem. A compact form of this Theorem will be used in an attempt to obtain Borell type inequalities for local times. Analogous to their recent work which shows the equivalence of sample path properties of the local times of a symmetric Markov process and an associated Gaussian process they will study the relationship between continuous additive functionals of a symmetric Markov process indexed by measures and an associated class of Gaussian chaoses indexed by measures. This will require an investigation into measure indexed Gaussian processes and chaoses in general. They will also try to extend their recent work on the law of the iterated logarithm for the local times of symmetric Levy processes and random walks to cover the interesting case of recurrent random walks on a two dimensional integer lattice. Professor Marcus and Rosen will continue their investigations of the relationship between Gaussian Processes and the local times of symmetric Markov processes. Until recently these stochastic processes were not considered to be related at all, except in the case of Brownian motion, a relationship was implied by a recent isomorphism theorem of Dynkin. This isomorphism has been exploited by Professors Marcus and Rosen to show that there are deep connections between these two fundamental classes of stochastic processes, both of which serve as models in countless applications to continuous time random phenomena. This research could lead to significant new knowledge about our random environment.