The Toda lattice, beyond being a completely integrable dynamical system, has many important properties.  Classically, the Toda flow is seen as acting on a specific class of bi-infinite Jacobi matrices.  Depending on the boundary conditions imposed for finite matrices, it is well known that the flow can be used as an eigenvalue algorithm. It was noticed by P. Deift, G. Menon and C. Pfrang that the fluctuations in the time it takes to compute eigenvalues of a random symmetric matrix with the Toda, QR and matrix sign algorithms are universal. In this talk, I will present a proof of such universality for the Toda algorithm.  I will also discuss empirical and rigorous results for other algorithms from numerical analysis.  This is joint work with P. Deift.

Boundary theory for one-dimensional diffusions is now well understood. Boundary theory for multi-dimensional diffusions is much richer and remains to be better understood. In this talk, we will be concerned with the construction and characterization of obliquely reflected Brownian motions in all bounded simply connected planar domains, including non-smooth domains,
with general reflection vector field on the boundary.  
We show that the family of all obliquely reflected Brownian motions in a given domain can be characterized in two different ways, either by the field of angles of oblique reflection on the boundary or by the stationary distribution and the rate of rotation of the process about a reference point in the domain. We further show that Brownian motion with darning and excursion reflected Brownian motion can be obtained as a limit of obliquely reflected Brownian motions.

Based on joint work with K. Burdzy, D. Marshall and K. Ramanan.

We consider a model of Brownian motion on a bounded interval which upon exiting the interval is being redistributed back  into the interval according to a probability measure depending on the exit point, then starting afresh, repeating the above mechanism indefinitely.  It is not hard to show that the process is exponentially ergodic, although characterizing the rate of convergence is non-trivial. In this talk, after providing a general overview of the probabilistic method of coupling and its applications,  I’ll show how to study the ergodicity for the model through coupling, how it leads to an  intuitive and geometric explanation for  the rates of convergence previously obtained analytically, other insights, and more questions. The talk will be accessible to general mathematical audience. 

  We consider the stochastic heat equation with a multiplicative space-time white noise forcing term under standard "intermitency conditions.” The main byproduct of this talk is that, under mild regularity hypotheses, the a.s.-boundedness of the solution$x\mapsto u(t\,,x)$ can be characterized generically by the decay rate, at $\pm\infty$, of the initial function $u_0$. More specifically, we prove that there are 3 generic boundedness regimes, depending on the numerical value of $\Lambda:=\lim_{|x|\to\infty} \vert\log u_0(x)\vert/(\log|x|)^{2/3}$.

In this talk, we consider kinetic equations containing random

terms. The kinetic models contain a small parameter and it is well

known that, after scaling, when this parameter goes to zero the limit

problem is a diffusion equation in the PDE sense, ie a parabolic equation

of second order. A smooth noise is added, accounting for external perturbation.

It scales also with the small parameter. It is expected that the limit

equation is then a stochastic parabolic equation where the noise is in

Stratonovitch form.

Our aim is to justify in this way several SPDEs commonly used.

We first treat linear equations with multiplicative noise. Then show how

to extend the methods to nonlinear equations or to the more physical

case of a random forcing term.

The results have been obtained jointly with S. De Moor and J. Vovelle.