Self-similarity and stable processes

随机过程（英语：Random process）是一种数学模型，描述一个在空间中随机移动的粒子随时间的演化。有许多不同的家庭的随机过程，现在，很好地理解不同程度的成功时，建立其他数学模型与应用在物理学，生物学和工程。其中一些更常用的随机过程是所谓的马尔可夫随机过程。对于这些过程，粒子在任何时刻的未来随机演化仅取决于其当前位置，而不取决于其迄今为止的历史路径。这个建议的目的是研究家庭的马尔可夫随机过程尊重自相似性的属性。粗略地说，随机过程是自相似的，当在空间和时间上进行适当的重新缩放后，所产生的随机轨迹是其自身的精确随机（分布）副本。这个提议的基本思想是试图理解一个自相似马尔可夫过程族（所谓的稳定过程）如何能够以异常的方式表现。这将通过研究嵌入稳定过程路径中的其他自相似结构来完成。特别是，我们感兴趣的是，如何上述分布条件否则可以被视为等价于在空间和时间的原始路径的进一步transformation.This有重要的分支的一般理解潜在的和随机分析的自相似马尔可夫过程，其中相对较少似乎是目前已知的相比，其他家庭的过程。这些知识，反过来，在一些应用概率models.The PI已经有一个大的EPSRC资助的项目正在进行中，在这个方向上，在这个建议的主要目标是扩大该机构的工作范围，以及加快其输出，通过资助一个12个月的访问，在这个领域的世界专家加入PI在巴斯的合作研究。

A stochastic process is a mathematical model for the evolution through time of a particle that moves randomly through space. There are many different families of stochastic processes that are, now, well understood with varying degrees of success when building other mathematical models with applications in physics, biology and engineering. Amongst some of the more commonly used stochastic processes are so-called Markov stochastic processes. For these processes, the future random evolution of the particle at any moment in time depends only on its current position and not on its historical path to date. This proposal aims to study families of Markov stochastic processes which respect the property of self-similarity. Roughly speaking, a stochastic process is self-similar when, after an appropriate re-scaling in space and time, the resulting random trajectory is an exact stochastic (distributional) copy of itself. The basic idea in this proposal is to try to understand how one family of self-similar Markov processes (so-called stable processes) can be conditioned to behave in an exceptional way. This will be done by studying other self-similar structures that are embedded in the path of the stable process. In particular, we are interested in how the aforementioned distributional conditioning can otherwise be seen as equivalent to further transformations in space and time of the original path. This has important ramifications for the general understanding of potential and stochastic analysis of such self-similar Markov processes, about which relatively little seems to be currently known in comparison to other families of processes. Such knowledge has, in turn, implications for the use of self-similarity in a number of applied probability models.The PI already has a large EPSRC-funded project underway in this direction and the main objective in this proposal is to expand the scope of that body of work, as well as accelerate its output, by funding a 12 month visit of a world expert in this field to join the PI in collaborative research in Bath.