Mathematics at Cambridge has a long and distinguished tradition. Today, there are groups of high international standing working in a large number of interrelated fields. Research in Stochastic Analysis takes place mainly within the Statistical Laboratory . The principal scientists working in this area are listed below, together with statements of some of their current research interests. In addition, there is a large and enthusiastic group of graduate students and post-docs.
Applications are welcome in any area of Stochastic Analysis for post-doctoral fellowships, funded jointly by the Isaac Newton Trust and the TMR Programme of the European Union. Further information is available on our local page. Intending applicants are encouraged to contact one or more of the individuals listed below to discuss the content of any post-doctoral proposal. Links are provided to personal home pages and to the Statistical Laboratory's online Research Reports series.
The following list will serve to indicate the range of topics in Stochastic Analysis and related areas currently under research at Cambridge: stochastic differential equations, stochastic partial differential equations, branching and measure-valued diffusions, large deviations, Dirichlet forms, analysis of heat kernels, rates of convergence for Markov processes, homogenization, stochastic analysis on manifolds in finite and infinite dimensions, Malliavin calculus, percolation, spatial processes, phase transition in random systems, mathematical finance, stochastic networks.
M.W. Baxter Works in the financial mathematics area by using stochastic analysis to formulate and analyse models of security and derivative markets. Current topics of importance are models of interest-rate markets, no-arbitrage conditions, and describing stochastic volatility with models of option prices.
D. Gatzouras Dimensional properties of attractors in dynamical systems, asymptotics of random Cantor sets. Here is a current problem: given a smooth expanding map f on a compact connected Riemannian manifold, there is always a (unique) ergodic f-invariant probability measure which is absolutely continuous with respect to Lebesgue measure. If we restrict f to a compact invariant set K, of zero Lebesgue measure, the analogue of this is an f-invariant probability measure which is supported by K and has the same Hausdorff dimension as K. Such measures are known to exist for smooth conformal functions f but otherwise little is known. This problem may also be connected with dimensional properties of self-affine sets and a better understanding of such properties is desirable.
G.R. Grimmett There is emphasis on percolation-type systems, such as classical percolation, the Ising/Potts models, and stochastic models for epidemics. Questions of homogenization in random media are relevant. A new interest is the rigorous study of mirror and rotator models, involving the analysis of a class of random walks through random fields of scatterers and reflectors.
D.P. Kennedy Work in mathematical finance has centred on modelling the term structure of interest rates as Gaussian random fields. Structural conditions such as different formulations of the Markov property and stationarity have been used to characterize possible forms of the covariance structure of these fields.
J.R. Norris 1. Stochastic partial differential equations of hyperbolic type provide a powerful tool in the analysis of path-valued processes in Riemannian manifolds. A programme to extend these techniques to loop-valued processes is currently underway. 2. Recent results give sharp global estimates on heat kernels with periodic coefficients, which are sharp in long-time. The analogous results for stochastically stationary coefficients are not known and call for further study.
G.O. Roberts Stochastic differential equations, Markov processes and quasi-stationarity, with applications in statistics and mathematical finance.
Y.M. Suhov Branching Markov processes, in particular, branching diffusion processes are studied, including processes on manifolds. Relations with linear and non-linear ordinary and partial differential equations are established and investigated.