Details of Grant 

EPSRC Reference:	GR/R63059/01
Title:	Functional integral methods in stochastic problems of fluid dynamics and quantum mechanics
Principal Investigator:	Truman, Professor A
Other Investigators:	Davies, Dr IM
Researcher Co-Investigators:
Project Partners:
Department:	Mathematics
Organisation:	Swansea University
Scheme:	Standard Research (Pre-FEC)
Starts:	24 October 2001	Ends:	23 October 2004	Value (£):	7,583
EPSRC Research Topic Classifications:	Mathematical Analysis
EPSRC Industrial Sector Classifications:	No relevance to Underpinning Sectors
Related Grants:
Panel History:
Summary on Grant Application Form
The propose research is concerned with continuing the development of functional integral methods in connection with stochastic problems in fluid mechanics and the theory of open quantum systems. The power of functional integral methods is undisputed in that they allow one to obtain representations of infinite dimensional pseudo-differential operators as integral operators . The main thrust is to be towards obtaining functional integral representations of solutions of stochastic Schrodinger type equations on Riemannian manifolds and on some infinite dimensional manifolds of trajectories in Riemannian manifolds. The investigation of stochastic Scrodinger equations for \Hamiltonian systems with constraints (Hamilton-Dirac systems) will form part of this aspect. Such method become applicable to problems in fluids, both classical and stochastic flavours, due to the close relationship between stochastic Schrodinger type equations, stochastic Hamilton-Jacobi equations and stochastic Burgers equations through the Hopf-Cole transformation. One may consider the physically important case when the fluids have non-trivial vorticity. It is worth emphasizing that the stochastic nature of the equations is due to general Levy white noise and not just to continuous noise. The connection with Nelson's stochastic mechanics, on manifolds, is not too difficult to make and we must emphasize the close connections with the viscous Burgers equation. The Nelson stochastic mechanics will be extended to include an approach based on p-representations. The latter part of the programme will concentrate on nonlinear transformations of Feynman measures in trajectories in Riemannian manifolds, a natural extension of previous work and the more topical are of computer simulation of solutions to heat, and other related, equations on manifolds. Such computer generated solutions can provide both motivation and guidance for further analytical developments. The visualization of solutions will be under consideration as will efficiency and speed.
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Organisation Website:	http://www.swan.ac.uk