Stationary solutions

Saturday, November 1, 2014 - 9:00am - 10:00am
Ivan Corwin (Columbia University)
Sinai and collaborators extensively studied questions related to stationary solutions to the Burgers equation driven by stochastic forcing. For space-time white noise forcing, it is known that 1d white noise is stationary. Equivalently, for the Kardar-Parisi-Zhang equation (integrated stochastic Burgers equation), two-sided Brownian motion is stationary, up to a height shift (given by the net current through the origin for the stochastic Burger equation).
Tuesday, December 4, 2012 - 2:00pm - 2:50pm
Bastien Fernandez (Centre National de la Recherche Scientifique (CNRS))
Bistable space-time discrete systems commonly possess a large variety of stable stationary solutions with
periodic profile. In this context, it is natural to ask about the fate of trajectories composed of interfaces
between steady configurations with periodic pattern and in particular, to study their propagation as
traveling fronts. In this talk, I will consider such fronts in piecewise affine bistable recursions on the one-dimensional lattice. By introducing a definition inspired by symbolic dynamics, I will present results on
Thursday, October 25, 2012 - 3:15pm - 4:05pm
Yuri Bakhtin (Georgia Institute of Technology)
The Burgers equation is a basic hydrodynamic model
describing the evolution of the velocity field of sticky dust
particles. When supplied with random forcing it turns into an
infinite-dimensional random dynamical system that has been studied
since late 1990's. The variational approach to Burgers equation allows
to study the system by analyzing optimal paths in the random landscape
generated by the random force potential. Therefore, this is
essentially a random media problem. For a long time only compact cases
Tuesday, September 25, 2012 - 10:15am - 11:05am
Yingfei Yi (Georgia Institute of Technology)
We consider white noise perturbations of a system of ordinary
differential equations. By relaxing the notion of Lyapunov functions associated with the stationary Fokker-Planck equations, new existence and non-existence
results of stationary measures in a general domain including the entire space
will be presented for both non-degenerate and degenerate noises. Limiting
behaviors of stationary measures will be discussed along with
applications to problems of stochastic stability and bifurcations.
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