Ian Davies : Research
Ian Davies : Research | |
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Track Record |  | |
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Recent work has seen a return to Poisson-Lévy excursion measures with a paper in the Electronic Journal of Probability documenting the full construction of the next to leading order term in the (asymptotic) expansion of the Poisson-Lévy excursion measure. It transpires that this term is identically zero and so we have the leading order term (having one of only two possible forms) giving an excellent approximation. One may extend the result, if sufficiently patient, as all necessary structures and techniques are discussed. A number of collaborations with colleagues in the School of Engineering, Prof Y Feng, Dr C Li and Prof R Owen FRS, have centered on the development of a mathematically rigorous theory of contimuum mechanics in stationary stochastic fields, a prototype analysis system for elasticity problems of random media. A Fourier-Karhunen-Loève discretization scheme is developed which exhibits a number of advantages over the widely used Karhunen-Loève expansion scheme based on Finite Element meshes, including better computational efficiency in terms of memory and CPU time, a convenient a priori error-control mechanism, better approximation accuracy of random material properties, explicit methods for predicting the associated eigenvalue decay speed and geometrical compatibility for random medium bodies of different shapes. Two papers have appeared, to date, with a couple still in draft format. An excursion of a different sort led to a small contribution to a paper appearing in “Deep Sea Research Part II: Topical Studies in Oceanography”. This paper considered methodological and analytical solutions for the resolution of movements built upon information recorded by data-logging devices. A short comparative study of discrete approximations to curvature and torsion followed and this should be of use in further studies. My work, with Aubrey Truman and Huaizhong Zhao, concerning the singularities of the stochastic heat and Burgers equations has been most fruitful. We have shown that a knowledge and deep understanding of the stochastic Hamilton principal function allows one to determine the nature of the caustics for the inviscid Burgers equation and the corresponding “wavefronts” for the stochastic heat equation. There are two preprints, and they may be downloaded from the Texas Mathematical Physics archive links Stochastic Heat and Burgers Equation, Stochastic Heat and Burgers Equation and their singularities II. Our first substantial paper in this area gives the detailed exposition for the geometrical properties. It appeared in Journal of Mathematical Physics 43, pp 3293--3328 in 2002 under the title Stochastic heat and Burgers equations and their singularities I, Geometrical properties. Our second paper appeared in Journal of Mathematical Physics 46 Stochastic heat and Burgers equations and their singularities II. Analytical properties and limiting distributions. We are currently preparing a further paper concerning the intermittence of turbulence. My interest in Gaussian Functional Integrals has been sustained over many years by the varied and useful applications that may be made of them. My most recent paper in this area appeared in the Electronic Journal of Probability and it presents what I believe to be the most useful exposition of the core problem. The paper combines ideas of Schilder, Pincus and the early work of Davies and Truman to give a general asymptotic expansion of a Gaussian functional integral up to any required order. I have studied the discrete random walk concentrating particularly on the time spent on links and nodes. There are several monographs which quote an “obvious” result and it was seeing this that led to a detailed study of the more general underlying system. This study has been extended to excursions and the opportunity exists to relate this work to its continuous analogues. There are connections with previous work of A Truman and D Williams and this is highlighted in a paper contained in the Gregynog Proceedings on Stochastic Analysis and Applications. I have studied the Poisson-Lévy excursion measure for a diffusion with small noise showing that there are only two forms for the leading order term: Lévy or Hawkes-Truman. I have studied the Kolmogorov, Petrovsky and Piscounoff equation. This reaction-diffusion equation is of great importance in modelling physical and biological systems. My motivation for this was a series of talks given by M.I. Freidlin at one of the Swansea meetings. The local work of A. Truman and H. Zhao has also been important in this respect. The numerical solution of the KPP equation for a variety of “classical” cases is important in that it enables one to develop an analytical understanding of the problem. Local experience with semi-classical mechanics was of great use in developing travelling wave solutions for this equation. After an initial period of study centred on the standard forms of the KPP equation I moved on to the stochastic partial differential equation. This was easy step to state but the techniques involved were in their infancy and great care had to be exercised in order to achieve any useful results. I performed a large number of numerical simulations in connection with the Stochastic KPP equation with multiplicative noise for many different types of noise. These simulations were of immense use in formulating the theoretical results presented in the joint paper of myself, A Truman and H Z Zhao. My main contribution in this area has been to derive a stochastic form of the Lagrangian action which allows one to obtain a Schrodinger equation without having to deal with divergent quantities. This method was extended to manifolds successfully and depends on the use of stochastic parallel translation. I have also written on the connection between observables in stochastic mechanics and their well known analogues in quantum mechanics. Another application of my earlier work on Conditional Wiener Integrals was in studying the Miessner-Ochsenfeld effect in superconductors. We took a gas of charged bosons in the presence of an homogeneous magnetic field and proved that after taking the thermodynamic limit the system would not support a magnetic field unless it were sufficiently strong. We also found that there was no macroscopic occupation of the ground state in the presence of a non-zero acting magnetic field. A short but immensely cited paper concerned the propagator for a harmonic oscillator and a constant magnetic field where the field is not in the direction of any of the eigenvectors of the quadratic form giving rise to the harmonic potential. This result was proved by using a good working knowledge of classical mechanics and Mehler Kernel formulae. The methods used therein continue to be of merit. A spinoff of my studies on quantum mechanical propagators was the discovery of new families of periodic solutions of the equations of motion for many body systems. The pairwise interactions may be viewed as Coulombic in nature and the methods used admit magnetic fields as well. This work led to two short papers appearing as well as one large working paper which has only had a limited circulation. My studies began with Laplace Asymptotic Expansions of Wiener and Conditional Wiener Integrals. Flat integral ideas and the seminal paper of Schilder lay at the heart of this study. Semi-classical methods were used to apply this powerful tool to several important problems including the Bender-Wu formula and the Charged Boson Gas | |
Recent Publications | | |
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MR2430708 I M Davies, Semiclassical analysis and a new result for Poisson-Lévy excursion measures, Electronic Journal of Probability 13, (2008), 1283 -- 1306 MR2402258 I M Davies, Y Feng, C F Li, D F Li, D R J Owen, A Fourier-Karhunen-Loève discretization scheme for stationary random material properties in SFEM, International Journal for Numerical Methods in Engineering 73, (2008), 1942 -- 1965 R P Wilson, N Liebsch, I M Davies, et al, All at sea with animal tracks: methodological and analytical solutions for the resolution of movement, Deep Sea Research Part II: Topical Studies in Oceanography 54, (2007), 193 -- 210 I M Davies, Y Feng, C F Li, D R J Owen, Fourier representation of random media fields in stochastic finite element modelling, Engineering Computations 23, (2006), 794 -- 817 MR2131267 I M Davies, A Truman, H Z Zhao, Stochastic heat and Burgers equations and their singularities II, Analytical Properties and Limiting Distributions, Journal of Mathematical Physics 46, (2005), no.4, 043515-1 -- 043515-31 MR2202034 I M Davies, A Truman and H Zhao, Stochastic heat and Burgers equations and their singularities, Probability and Partial Differential Equations in Modern Applied Mathematics, J Duan and E C Waymire (eds), The IMA Volumes in Mathematics and its Applications 140, (New York, Springer Verlag, 2005), 79 -- 96, ISBN: 0-387-25879-5 Complete publications list. | |