# 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)
Boundary control for stochastic Navier-Stokes equations.

Participating Groups: