We consider Burgers type nonlinear SPDEs with L
We follow Arnold's approach of Euler equation as a geodesic on the group of
diffeomorphisms. We construct a geometrical Brownian motion on this group in the
case of the two dimensional torus, and prove the global existence of a
stochastic perturbation of Euler equation (joint work with F. Flandoli and P.
Malliavin).
Other diffusions allow us to obtain the deterministic Navier-Stokes equation
as a solution of a variational problem (joint work with F. Cipriano).
In this talk, the convergence analysis of a class of weak approximations of
solutions of stochastic differential equations is presented. This class includes
recent approximations such as Kusuoka's moment similar families method and the
Lyons-Victoir cubature on Wiener Space approach. It will be shown that the rate
of convergence depends intrinsically on the smoothness of the chosen test
function. For smooth functions (the required degree of smoothness depends on the
order of the approximation), an equidistant partition of the time interval on
which the approximation is sought is optimal. For functions that are less smooth
(for example Lipschitz functions), the rate of convergence decays and the
optimal partition is no longer equidistant. An asymptotic rate of convergence
will also be presented for the Lyons-Victoir method. The analysis rests upon
Kusuoka-Stroock's results on the smoothness of the distribution of the solution
of a stochastic differential equation. Finally, the results will be applied to
the numerical solution of the filtering problem.
We consider multi-dimensional Gaussian processes and give a novel, simple and
sharp condition on its covariance (finiteness of its two dimensional rho-variation,
for some rho <2) for the existence of "natural" Levy areas and higher iterated
integrals, and subsequently the existence of Gaussian rough paths. We prove a
variety of (weak and strong) approximation results, large deviations, and
support description.
Rough path theory then gives a theory of differential equations driven by
Gaussian signals with a variety of novel continuity properties, large deviation
estimates and support descriptions generalizing classical results of
Freidlin-Wentzell and Stroock-Varadhan respectively.
(Joint work with Nicolas Victoir.)