# Workshop 1: Randomness and Partial Differential Equations

###List of (confirmed) speakers

Cédric Bernardin, ÉNS Lyon
Arnaud Debussche, ENS Bretagne et IRMAR
Arnaud Guillin, Université de Clermont-Ferrand
Tony Lelièvre, ÉNPC-Paris Est
Jan Maas, University of Bonn
Laurent Miclo, Université of Toulouse
Yann Ollivier, Université Paris-Sud
Grigorios Pavliotis, Imperial College Londres
Laurent Thomann, Université de Nantes
Vincent Vargas, Université Paris Dauphine

###Titles and abstracts
• Cedric Bernardin (ENS Lyon) : *Anomalous diffusion in Hamiltonian systems perturbed by a conservative noise* I will discuss a class of Hamiltonian systems perturbed by a conservative noise in the spirit of models considered in Basile-Bernardin-Olla '06-'09. For exponential interactions I will show that the system is super diffusive. This is based on joint works with P. Goncalves and G. Stoltz.

• Arnaud Debussche (ENS Bretagne et IRMAR) : *Existence of densities for stochastic equations with non smooth coefficients.* In this talk, we present a new method to prove existence of densities. It has the advantage that it does not use Malliavin calculus and thus can be applied to more general equations. On another hand, the obtained densities have low regularity. They are in the Besov spaces $B^{s}_{1,\infty}$ with $s$ positive but small. We apply this method to the stochastic Navier-Stokes equations in dimension $3$ and to Levy driven stochastic differential equations with H\"older coefficients. These results are obtained in collaboration with N. Fournier and M. Romito.

• Arnaud Guillin (Clermont Ferrand) : *Inegalites locales pour des diffusions non reversibles.* Pour demontrer des inegalites locales, la methode usuelle repose sur le critere de Bakry-Emery, une commutation entre le Carre du champ et le semi groupe, et un calcul analytique astucieux. Nous tenterons de montrer ici une methode directe, basee sur des controles en distance de Wasserstein, du couplage et des h-processus, qui s'applique de plus dans des cas non reversibles, hypoelliptiques ou la courbure n'est pas calculable, ou infinie negative. (en collab. avec P. Cattiaux)

• Tony Lelièvre (Paris, Cermics) : *Optimal scaling of the transient phase of Metropolis Hastings algorithms* We consider the Random Walk Metropolis algorithm on R^n with Gaussian proposals, and when the target probability measure is the n-fold product of a one dimensional law. It is well-known that, in the limit n goes to infinity, starting at equilibrium and for an appropriate scaling of the variance and of the timescale as a function of the dimension n, a diffusive limit is obtained for each component of the Markov chain. We generalize this result when the initial distribution is not the target probability measure. The obtained diffusive limit is the solution to a stochastic differential equation nonlinear in the sense of McKean. We prove convergence to equilibrium for this equation using entropy techniques. We discuss practical counterparts in order to optimize the variance of the proposal distribution to accelerate convergence to equilibrium. Our analysis confirms the interest of the constant acceptance rate strategy (with acceptance rate between 1/4 and 1/3). This is a joint work with B. Jourdain and B. Miasojedow.

• Jan Maas (Bonn) : *Approximating rough stochastic PDEs* We consider Burgers type stochastic PDEs in one space dimension driven by space-time white noise. Although these equations are perfectly well-posed, they turn out to be very unstable under approximation of the nonlinearity. In fact, we prove that solutions to different - seemingly natural - approximating schemes converge to different limits. This phenomenon can be understood in the framework of rough paths theory. This is joint work with Martin Hairer and Hendrik Weber (Warwick).

• Laurent Miclo (Toulouse III) : *On hyperboundedness and spectrum of Markov operators.* Consider an ergodic Markov operator $M$ reversible with respect to a probability measure $\mu$ on a general measurable space.We will show that if M is bounded from L^2(mu) to L^p(\mu), where p>2, then it admits a spectral gap. This result answers positively a conjecture raised by Hoegh-Krohn and Simon in a semi-group context. The proof is based on isoperimetric considerations and especially on Cheeger inequalities of higher order for weighted finite graphs recently obtained by Lee, Gharan and Trevisan.In general there is no quantitative link between hyperboundedness and spectral gap (except in the situation investigated by Wang), but there is one with another eigenvalue. In addition, the usual Cheeger and Buser inequalities will be extended to higher eigenvalues in the compact Riemannian setting.

• Yann Ollivier (Paris, Orsay) : *Curvature of Markov chains and processes, with applications.* The notion of Ricci curvature of a Markov chain or process on a metric space, which can be traced back to Dobrushin in the 1960's and connects it to Riemannian geometry, is a useful tool for proving convergence results for the process as well as various concentration and functional inequalities. We will present this criterion and focus on its applications. These range from new and sometimes near-optimal concentration inequalities for Monte Carlo Markov chain simulation techniques (work with Alderic Joulin), such as waiting queues, to improvement in classical estimates for differential geometry, such as the spectral gap of the Laplace-Beltrami operator, to spectral gap estimates for some semi-elliptic diffusions (work of Laurent Veysseire).

• Grigorios Pavliotis (Londres, Imperial College) : *Convergence to equilibrium for nonreversible diffusions.* The problem of convergence to equilibrium for diffusion processes is of theoretical as well as applied interest, for example in nonequilibrium statistical mechanics and in statistics, in particular in the study of Markov Chain Monte Carlo (MCMC) algorithms. If we want to sample from a probability distribution, then we can construct many different ergodic diffusion processes whose invariant distribution is the one from which we want to sample. Our objective is to choose the diffusion process that converges the fastest to equilibrium. It turns out that the addition of a nonreversible perturbation to the reversible overdamped Langevin dynamics can accelerate convergence to equilibrium. In this work we show that for diffusions with a linear drift we can calculate the optimal convergence rate and we develop an algorithm for obtaining the optimal nonreversible perturbation. Our theoretical findings are supported by numerical experiments. This is joint work with Tony Lelievre and Francis Nier.

• Laurent Thomann (Nantes) : *Nonlinear Schrödinger equation and random initial conditions.* Some nonlinear Schrödinger equations are known to be ill-posed in some irregular spaces. However, using random initial conditions, we will see that we can sometimes almost surely solve the Cauchy problem in these situations. The results are based on collaborations with N. Burq, N. Tzvetkov, A. Poiret and D. Robert.

• Vincent Vargas (Paris Dauphine) : *Liouville Brownian motion.* I will introduce a Feller process, called the Liouville Brownian motion (LBM), which is conjectured to be the scaling limit of random walks on large planar maps which are embedded in the Euclidean plane or in the sphere in a conformal manner. In the first part of this talk, I will give the main steps of its construction and review the theory of Gaussian multiplicative chaos (which is crucial in the construction of the LBM and enables to construct the associated reversible measure). In the second part, I will study the associated Dirichlet form and prove the existence of the associated heat kernel, the Liouville heat kernel. Based on joint work with C. Garban and R. Rhodes.