Self-similarity and stable processes

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.

The bigger picture:
Applied probability has seen huge waves of success thanks to the introduction of new mathematically robust families of Markov processes and their theories; to name but two concrete examples, we may consider one-dimensional diffusions and processes with stationary and independent increments. For more than half a century, the latter have found a firm footing in many different pure and applied fields of science and social science (e.g. physical models, engineering science, population biology and genetics, modelling of the atmosphere and weather systems, to name but a few), but there is still a desire for more competitive models, which are capable of harbouring ever more realistic qualitative and quantitative features of pertinence. Self-similarity (in combination with Markovian structure) is a highly desirable property of many physical, biological and economical systems. With the foundational theory of real-valued self-similar Markov processes in hand, the exploration of how self-similarity may be built into many different kinds of applied probability models may begin. This proposal fits in at the very beginning of that process and, in combination with other funded research activities of the PI, will help establish the robustness of the new theory surrounding rssMp with a view to ongoing applications.

Training opportunities:
This proposal is concerned with bringing Dr. Rivero from CIMAT Mexico to Prob-L@B the University of Bath for a 12-month visit. He is a recognised world expert in the field of self-similar Markov processes as well as having broader expertise in potential analysis of Markov processes. Prob-L@B is one of the largest and most vibrant research groups focused on probability theory both nationally and internationally. With an average portfolio of 12 Ph.D. students and 4 postdoctoral researchers and a well organised platform supporting interactive research training, there can be no doubt that the addition of Dr. Rivero to the existing strength of expertise supporting this environment will have a positive effect. Care has been taken to make sure that this effect is amplified by ensuring that he gives 20-weeks of advanced research lectures (10 weeks in the first semester on the topic of potential analysis, 10 weeks in the second semester on the topic of self-similarity). This is done deliberately in parallel to the developments of the research programme of the PI in self-similarity to generate interest from young probabilists, thereby ensuring that the UK maintains a competitive advantage in this field of research.

Internationalisation:
Globalisation, the subsequent generation of wealth and better standards of living has meant that a number of countries have recently experienced rapid state investment in their respective academic systems. Accordingly, a number of new competitive players have emerged in the global sphere of intellectual schools of thought. In the case of probability theory, for various reasons, there has been a surge of exceptionally high quality expertise coalescing at two institutes in Mexico: CIMAT and UNAM. It is important part of Prob-L@B's mandate to engage as much as possible with other international centres of excellence with a view to supporting and encouraging the highest quality and most relevant research output possible. This bears particular relevance to the future strength of the UK's probability research community as it is now the case that a recognisable proportion of young UK-based probabilists are trained in Prob-L@B. An excellent and practical example of what is meant here is the case of one of the PI's recent Ph.D. students who went on to take up a postdoctoral research position in CIMAT in 2013, before moving on to a second postdoctoral position with one of the world's leading probabilists in Zurich in 2014. In due course this young researcher will return to the UK with a highly competitive CV ready contribute to the local academic market