Self-similarity and stable processes

随机过程是一个粒子在空间中随机运动的数学模型。现在，在物理学、生物学和工程学中建立其他数学模型时，人们已经很好地理解了许多不同种类的随机过程，并取得了不同程度的成功。其中一些更常用的随机过程是所谓的马尔科夫随机过程。对于这些过程，粒子在任何时刻的未来随机演化只取决于它当前的位置，而不取决于它迄今为止的历史路径。本文旨在研究尊重自相似性质的马尔可夫随机过程族。粗略地说，随机过程是自相似的，当在空间和时间上适当地重新缩放后，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组自相似的马尔可夫过程（所谓的稳定过程）是如何以一种特殊的方式被调节的。这将通过研究嵌入在稳定过程路径中的其他自相似结构来完成。特别地，我们感兴趣的是如何将前面提到的分布条件条件看作是原始路径在空间和时间上的进一步转换。这对于一般理解这种自相似马尔可夫过程的潜在和随机分析具有重要的影响，与其他过程族相比，目前对这种过程的了解相对较少。这些知识反过来又暗示了在一些应用概率模型中使用自相似性。PI已经在这个方向上有一个由epsrc资助的大型项目正在进行中，该提案的主要目标是通过资助该领域的一位世界专家在巴斯进行为期12个月的访问来加入PI的合作研究，从而扩大该工作主体的范围，并加速其产出。

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.