# SETSAC York University

**Zdzislaw Brzezniak**

**1.**Numerical approximation of the stochastic Navier-Stokes equations, in particular:

(i) approximation of the invariant measure,

(ii) splitting method for the stochastic Navier-Stokes equations,

**2.** Splitting method for the stochastic nonlinear Schroedinger equation.

**3.** Stochastic wave (and similar) equations with values in Riemannian manifolds.

**4.** Study of the invariant measures of deterministic partial differential equations and of random dynamical systems described by SPDEs (like Navier-Stokes, wave, Ginzburg-Landau and Cahn-Hilliard equations) and their small noise asymptotics. In particular, find a characterization of physically relevant invariant measures and their properties.

**Nigel Cutland**

**1.**Further work on Loeb space methods for stochastic Navier-Stokes and related equations:

a. having found solutions to the stochastic incompressible non-homogeneous equations issues to be investigated include attractors and optimal control theory

b. extension of the Loeb space techniques to compressible stochastic Navier-Stokes equations

c. extension of the techniques to unbounded domains

**2.** Work on rough paths theory:

a. trying to see the connection between Lyons' pathwise stochastic integration for rough paths and the nonstandard approach to stochastic integration which is also in a sense pathwise (actually $\omega$- wise.

b. developing an infinitesimal (i.e. nonstandard) approach to Lyons' theory - in particular to enable a difference equation approach to the solution of rough path driven DEs and provide a more natural solution concept (and the existence of solutions). This could be related to Euler schemes for solution of such equations.

**Rafael Perdomo (PhD Student of Zdzislaw Brzezniak)**