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Random matrices and stochastic analysis
Areas of research interest of Gordon Blower and James Groves.
Random matrices
Random matrices were considered by statisticians in the 1930s, but their systematic study began later in applications to nuclear physics. Wigner observed that the energy levels of a nucleus with n interacting nucleons could be modelled well by the eigenvelues of an n by n matrix, for which the (j,k)th entry represented the interaction between the nucleons j and k. Random matrices now feature in models of many interacting particle systems, as developed by Dyson and other over several decades.
In the last ten years Pastur and co-authors have developed the mean field approach to the spectral theory of random matrices, which applies to ensembles that are inaccessible to the orthogonal polynomial technique that is described in Mehta's famous book. The probabilistic properties of these ensembles are well described by the concentration of measure phenomenon. Gordon Blower has proved results about the convergence of eigenvalue distributions using modern convexity theory.
- G. Blower, Almost sure weak convergence for the generalized orthogonal ensemble, Journal of Statistical Physics, 105 (2001), 309-335. JSP
- G. Blower, Displacement convexity for the generalized orthogonal ensemble, Journal of Statistical Physics, 116 (2004) no. 5/6, 1359-1387. JSP
Semigroups
The theory of semigroups is a powerful tool in stochastic analysis; that is, the study of shochastic processes in continuous time. Of fundamental importance are stationary processes such as the Ornstein-Uhlenbeck process; this describes the motion of particles in a resistive medium which experience random collisions. James Groves (former PhD student) has analysed versions of the OU process that evolve in suitable Banach spaces and that are governed by stochastic differnetial equations involving semigroup generators.
Gordon Blower has developed a theory of maximal operators for semigroups. The maximal theorems allow one to prove almost sure convergence to the initial data of solutions to Cauchy problems, when the initial data are mildly smooth. Whereas the hypotheses of these theorems are specific, the proofs involve both abstract results from the functional calculus of operator groups, and properties of Mellin transforms.
