C6

When studying a systems of conservation laws in several space dimensions, A. Bressan conjectured that the flows $X^n(t)$ generated by a smooth vector fields $\mathbf b^n(t,x)$,
\[
\frac{d}{dt} X^n(t,y) = \mathbf b^n(t,X(t,y)),
\]
are compact in $L^1(I\!\!R^d)$ for all $t \in [0,T]$ if $\mathbf b^n \in L^\infty \cap \mathrm{BV}((0,t) \times I\!\!R^d)$ and they are nearly incompressible, i.e.
\[
\frac{1}{C} \leq \det(\nabla_y X(t,y)) \leq C
\]
for some constant $C$. This conjecture is implied by the uniqueness of the solution to the linear transport equation
\[
\partial_t \rho + \mathrm{div}_x(\rho \mathbf b) = 0, \quad \rho \in L^\infty((0,T) \times I\!\!R^d),
\]
and it is the natural extension of a series of results concerning vector fields $\mathbf b(t,x)$ with Sobolev regularity.

We will give a general framework to approach the uniqueness problem for the linear transport equation and to prove Bressan's conjecture.

Abstract: In 2016 Ayyer, Prasad and Spallone proved that the restriction to 
S_{n-1} of any odd degree irreducible character of S_n has a unique irreducible 
constituent of odd degree.
This result was later generalized by Isaacs, Navarro Olsson and Tiep.
In this talk I will survey some recent developments on this topic.

We consider families of fast-slow skew product maps of the form \begin{align*}x_{n+1}   = x_n+\eps^2 a_\eps(x_n,y_n)+\eps b_\eps(x_n)v_\eps(y_n), \quad

y_{n+1}   = T_\eps y_n, \end{align*} where $T_\eps$ is a family of nonuniformly expanding maps, $v_\eps$ is of mean zero and the slow variables $x_n$ lie in $\R^d$.  Under an exactness assumption on $b_\eps$ (automatically satisfied in the cases $d=1$ and $b_\eps\equiv I_d$), we prove convergence of the slow variables to a limiting stochastic differential equation (SDE) as $\eps\to0$.   Our results include cases where the family of fast dynamical systems

$T_\eps$ consists of intermittent maps, unimodal maps (along the Collet-Eckmann parameters) and Viana maps.Similar results are obtained also for continuous time systems  \begin{align*} \dot x   =  \eps^2 a_\eps(x,y,\eps)+\eps b_\eps(x)v_\eps(y), \quad \dot y   =  g_\eps(y). \end{align*}

Here, as in classical Wong-Zakai approximation, the limiting SDE is of Stratonovich type $dX=\bar a(X)\,dt+b_0(X)\circ\,dW$ where $\bar a$ is the average of $a_0$

and $W$ is a $d$-dimensional Brownian motion.

Singular stochastic control problems ae largely studied in literature.The standard approach is to study the associated Hamilton-Jacobi-Bellman equation (with gradient constraint). In this work, we use a different approach (BSDE:Backward stochastic differntial equation approach) to show that the optimal value is a solution to BSDE.

The advantage of our approach is that we can study this kind of singular stochastic control with path-dependent coefficients

We consider fully nonlinear parabolic equations of the form $du = F(t,x,u,Du,D^2 u) dt + H(x,Du) \circ dB_t,$ which can be made sense of by the Lions-Souganidis theory of stochastic viscosity solutions. I will first recall the ideas of this theory, and will discuss more recent developments (including the use of rough path theory in this context). In the second part of my talk, I will explain how in the case where $H(x,Du)=|Du|^2$, the solution $u$ may enjoy better regularity properties than the solution to the unperturbed equation, which can be measured by (a pair of) solutions to a reflected SDE. Based on joint works with P. Friz, B. Gess, P.L. Lions and P. Souganidis.

The planar Ising model is one of the simplest and most studied models in Statistical Mechanics. On one hand, it has a rich and interesting phase transition behaviour. On the other hand, it is "solvable" enough to allow for many rigorous and exact results. This, in particular, makes it one of the prime examples in Conformal Field Theory (CFT). In this talk, I will review my joint work with C. Hongler and D. Chelkak on the scaling limits of correlations in the planar Ising model at criticality. We prove that these limits exist, are conformally covariant and given by explicit formulae consistent with the CFT predictions. This may be viewed as a step towards a rigorous understanding of CFT in the case of the Ising model.TBC

We consider the random conductance model: random walk among iid, uniformly elliptic conductnace on the d-dimensional lattice. We state,and explain, the Einstein relation for this model:It says that the derivative of the velocity of a biased walk as a function of the bias equals the diffusivity in equilibrium. For fixed bias, we show that there is an invariant measure for the environment seen from the particle.These invariant measures are often called steady states.

The Einstein relation follows, at least for dimensions three and larger, from an expansion of the steady states as a function of the bias.

The talk is gase on joint work with Jan Nagel and Xiaoqin Guo

This talk will be about  Lévy processes on compact groups - discrete or continuous - and  two-dimensional analogues called pure Yang-Mills fields. The latter are indexed by  reduced loops of finite length in the plane and satisfy properties analogue to independence and stationarity of increments.     There is a one-to-one correspondance between Lévy processes invariant by adjunction and pure Yang-Mills fields. For Brownian motions, Yang-Mills fields stand for a rigorous version of the Euclidean Yang-Mills measure in two dimension.  I shall first sketch this correspondance for  Lévy processes with large jumps. Then, I will discuss two applications of an extension theorem, due to Thierry Lévy, similar to Kolmogorov extension theorem. On the one hand, it allows to construct pure Yang-Mills fields for any invariant Lévy process. On the other hand, when the group acts on vector spaces of large dimension, this theorem also allows to study the asymptotic behavior  of traces. The limiting objects yield a natural family of states on the group algebra of reduced loops.  We characterize among them the master field defined by Thierry Lévy by a continuity property.   This is  a joint work with Guillaume Cébron and Franck Gabriel.