The seminar meets regularly on wednesday at 4pm in Eckhart 202. We also have special seminars during other days.
The KPZ equation was originally introduced in the eighties as a model for surface growth, but it was soon realised that its solution is a "universal" object describing the crossover between the Gaussian universality class and the KPZ universality class. The mathematical proof of its universality however is still an open problem, in particular because of the lack of a good approximation theory for the equation. In this talk, we present a new notion of solution to the KPZ equation that bypasses the use of the Cole-Hopf transform. One key feature of our theory is that it exhibits a topology on the noise space which is such that the solution is continuous as a function of both the initial condition and the driving noise. This lays the foundations for a robust approximation theory to the KPZ equation, which is needed to prove its universality. As a byproduct of the construction, we obtain very detailed regularity estimates on the solutions, as well as a new homogenisation result.
The classical way of measuring the regularity of a function is by comparing it in a neighbourhood of any point with a polynomial of sufficiently high degree. Would it be possible to replace monomials by functions with less regular behaviour or even by distributions? It turns out that the answer to this question has surprisingly far-reaching consequences for building solution theories for semilinear PDEs with very rough input signals, revisiting the age-old problem of multiplying distributions of negative order, and understanding renormalisation theory.
We will present recent and ongoing works, in collaboration with Tony Lelievre and other colleagues, that all aim at mathematically formalizing various techniques introduced about a decade ago by A. Voter (Los Alamos) to accelerate molecular dynamics techniques. Exploring a physical energy landscape (configuration or phase space) using a molecular dynamics type approach is a challenging issue because the dynamics is a succession of long stays in metastable states and rapid transitions between them. Accelerating the dynamics is thus mandatory for the numerical practice in order to eventually visit all attainable states. It is a challenging issue per se. But an even more challenging question is to also recover, behind the artificially accelerated dynamics, the underlying physical dynamics, in terms of actual sequence of states visited and actual time needed to visit them. We will describe some approaches, and the first elements of mathematical formalizations of them. A number of open theoretical and numerical issues will be mentioned. Reference: A mathematical formalization of the parallel replica dynamics, C. Le Bris, T. Lelievre, M. Luskin, D. Perez, published in Monte-Carlo Methods and Applications, available on Arxiv .
I will discuss a new version of the PDE approach, introduced by G. Barles and P. E. Souganidis in 2000, to the large time asymptotic behavior of solutions of Hamilton-Jacobi equations as well as key assumptions in the main result. The talk will be based on a joint work with G. Barles and H. Mitake.
I will talk on the study which is based on the recent joint work with Hung V. Tran from the University of Chicago. In this talk, we will discuss on the large-time behavior of the value functions of the optimal control problems on the n-dimensional torus which appear in the dynamic programming for the system whose states are governed by random changes.