The study of SPDEs by solving Sobolev space valued solution of BDSDEs. The SPDEs covered are very general, including a type of SPSEs with noise term nonlinearly depending on the solution and its gradients, the second order differential operator can be degenerate. Moreover, the stationary solution of SPDEs is given by the solution of the BDSDEs in infinite horizon. Implications for the numerics of SPDEs that can be transformed to the numerical solution of BDSDEs.
SPDEs as stochastic dynamical systems. The existence of stochastic flows and the stable/unstable manifolds near a pathwise stationary solution solved for SPDEs and stochastic evolution equations. (In collaboration with S.Mohammed (Carbondale) and T.Zhang (Manchester))
The existence of a pathwise stationary solution of stochastic systems. Due to the fact that the external random force is pumped to the system constantly, such a pathwise stationary solution (random fixed point) consists of infinitely many random moving invariant surfaces on the configuration space and therefore is a more realistic model than deterministic systems in many physical problems. However, in contrast to the deterministic systems, the existence of stationary solutions of stochastic dynamical systems is a difficult and subtle problem.
Extension of these results to: stochastic Burgers equations, Navier-Stokes equations arising in fluid mechanics, the AMLE model arising in imagining processing, and random periodic solutions, are being investigated by Zhao and his group in Loughborough.
Mr. Xiaohui Yan (Loughborough)
Mr. Andrei Yevik (Loughborough)
Miss Peng Lian (Loughborough)