• Thematic program: Random Phenomena in Partial Differential Equations (2007)

We derive the precise limit of SHS in the high activation energy scaling suggested by B.J. Matkowksy-G.I. Sivashinsky in 1978 and by A. Bayliss-B.J. Matkowksy-A.P. Aldushin in 2002. In the time-increasing case the limit coincides with the Stefan problem for supercooled water {\em with spatially inhomogeneous coefficients}. In general it is a nonlinear forward-backward parabolic equation {\em with discontinuous hysteresis term}. In the first part of the talk we give a complete characterization of the limit problem in the case of one space dimension. In the second part we construct in any finite dimension a rather large family of pulsating waves for the limit problem. In the third part, we prove that for constant coefficients the limit problem in any finite dimension {\em does not admit non-trivial pulsating waves}. The combination of all three parts strongly suggests a relation between the pulsating waves constructed in the present paper and the numerically observed pulsating waves for finite activation energy in dimension $n\ge 1$ and therefore provides a possible and surprising explanation for the phenomena observed.

• Thematic program: Random Phenomena in Partial Differential Equations (2007)

We study approximation strategies for the limit problem arising in the homogenization of Hamilton-Jacobi equations. They involve first an approximation of the effective Hamiltonian then a discretization of the Hamilton-Jacobi equation with the approximate effective Hamiltonian. We give a global error estimate which takes into account all the parameters involved in the approximation.

• Thematic program: Random Phenomena in Partial Differential Equations (2007)
• Event: Workshop "Stochastic Problems and Degenerate Elliptic Equations" (2007)

There are many references showing that a classical solution to the Black-Scholes equation is a stochastic solution. However, it is the converse of this theorem that is most relevant in applications, and the converse is also more mathematically interesting. In this talk we establish such a converse. We find a Feynman-Kac-type theorem showing that the stochastic representation yields a classical solution to the corresponding Black-Scholes equation with appropriate boundary conditions under very general conditions on the coefficients. We also study the pricing equation in the presence of bubbles, ie when the price process is a strict local martingale. In this case there is an infinite dimensional space of classical solutions. These results are obtained jointly with Svante Janson and Erik Ekström, respectively.

• Thematic program: Random Phenomena in Partial Differential Equations (2007)
• Event: Workshop "Stochastic Problems and Degenerate Elliptic Equations" (2007)

Let us consider 3D Navier-Stokes (NS) equations perturbed by a degenerate random force. A solution $u(t,x)$ of this problem is a random process in an appropriate functional space. We say that the solution $u$ is stationary if the law of $u(t,cdot)$ does not depend on time. A stationary measure for the NS equations is defined as the law of a stationary solution. The aim of my talk is to present some qualitative properties of stationary measures. Roughly speaking, we show that if the random perturbation is sufficiently non-degenerate, then the support of any stationary measure coincides with the entire phase space, and its finite-dimensional projections are minorised by the Lebesgue measure multiplied by a smooth positive density.

• Thematic program: Random Phenomena in Partial Differential Equations (2007)
• Event: Workshop "Stochastic Problems and Degenerate Elliptic Equations" (2007)

We survey a general approach to singular perturbations and homogenization problems for Hamilton-Jacobi-Bellman-Isaacs 1st and 2nd order equations arising in the reduction of dimension of multiscale control systems. They are formulated for optimal stochastic control problems or for zero-sum differential games, via the associated dynamic programming PDEs and their viscosity solutions. In particular, we present results for problems with an arbitrary number of scales and with oscillating terms in the PDE as well as in the initial data. Most of the results are obtained in collaboration with O. Alvarez and C. Marchi.

• Thematic program: Random Phenomena in Partial Differential Equations (2007)
• Event: Workshop "Stochastic Problems and Degenerate Elliptic Equations" (2007)

We show that a natural class of hypo-elliptic processes on Lie groups admits an invariant measure and a spectral gap with respect to it. We apply this class of processes to construct simulated annealing algorithms which converge in distribution to minima of non-convex functionals. The algorithms are non-elliptic and need therefore less independent Brownian motions than space dimensions. The universal constants depend on the geometry of certain nilpotent Lie groups. We apply the Driver-Melcher inequalities on Lie groups to show the main estimates.

• Thematic program: Random Phenomena in Partial Differential Equations (2007)
• Event: Workshop "Stochastic Problems and Degenerate Elliptic Equations" (2007)

We know how to prove an homogenization result, by a probabilistic method, for the solution $u^\eps$ of an elliptic or parabolic second order PDE with periodic coefficients, even when we allow the matrix of second order coefficients to degenerate, for example to vanish on an open set. In this talk, we will concentrate on the caracterization of the range of the homogenized diffusion matrix (in particular we shall say when this matrix is non degenerate). The results are joint with Martin Hairer (Warwick).

• Thematic program: Random Phenomena in Partial Differential Equations (2007)
• Event: Workshop "Stochastic Problems and Degenerate Elliptic Equations" (2007)

I will review some recent results on existence of viscosity solutions to systems of variational inequalities with inter-connected obstacles, driven by a second order linear operator. We give an equivalent formulation as an optimal multi-switching problem, whose solution is given by solving a system of reflected backward SDEs with oblique reflection. This is joint work with S. Hamadéne.

• Thematic program: Random Phenomena in Partial Differential Equations (2007)
• Event: Workshop "Stochastic Problems and Degenerate Elliptic Equations" (2007)

In this talk we describe the generalized Mather problem and its connections with stochastic optimal control. Namely, we will establish representation formulas for viscosity solutions and show how these formulas imply uniqueness of solutions.

• Thematic program: Random Phenomena in Partial Differential Equations (2007)
• Event: Workshop "Stochastic Problems and Degenerate Elliptic Equations" (2007)