UBC Mathematics Colloquium
Tadahiro Oh
(University of Toronto)
Probability in the PDE theory
Wed., Feb. 3, 2010, 3:00pm, WMAX 110
Abstract:
In this talk, we discuss how probabilistic ideas are applied to study
PDEs. First, we briefly go over the basic theory of Gaussian Hilbert
spaces and abstract Wiener spaces to determine function spaces which
capture the regularity of the Brownian motion and the white
noise.
Next, we go over Bourgain's idea to establish the invariance of Gibbs
measures for PDEs. We then establish local well-posedness (LWP) of KdV
with the white noise as initial data via the second iteration
introduced by Bourgain. This in turn provides almost sure global
well-posedness (GWP) of KdV as well as the invariance of the white
noise. Then, we discuss how one can use the same idea to obtain LWP of
the stochastic KdV with additive space-time (non-smoothed) white noise
in the periodic setting.
We also consider the weak convergence problem of the grand
canonical
ensemble (i.e. the interpolation measure of the usual Gibbs measure and
the white noise) with a small parameter (tending to 0) to the white
noise. This result, combined with the GWP in $H^{-1}$ by Kappeler and
Topalov, provides another proof of the invariance of the white noise
for KdV. In this talk, we discuss the same weak convergence problem for
mKdV and cubic NLS, which provides the ``formal'' invariance of the
white noise. This part is a joint work with J. Quastel and B. Valk\'o.
Lastly, if time permits, we discuss the well-posedness of the Wick
ordered cubic NLS on the Gaussian ensembles below $L^2$. The main
ingredient is nonlinear smoothing under randomization of initial data.
For GWP, we also use the invariance (of the Gaussian ensemble) under
the linear flow. This part is a joint work with J. Colliander.