Programme

Program of Workshop 
Functional Inequalities and SPDES II
Warsaw June 4 – 6, 2018
Banach Center, room 321.

List of Speakers:

Monday, 4.06.2018 

9:00-9:50, A. Novikov (Penn. State Univ.)  A fractional kinetic process describing the intermediate time behaviour of cellular flows.

Abstract: This is joint work with Martin Hairer, Gautam Iyer, Leonid Koralov, and Zsolt Pajor-Gyulai. This work studies the intermediate time behaviour of a small random perturbation of a periodic cellular flow. Our main result shows that on time scales shorter than the diffusive time scale, the limiting behaviour of trajectories that start close enough to cell boundaries is a fractional kinetic process: A Brownian motion time changed by the local time of an independent Brownian motion. Our proof uses the Freidlin-Wentzell framework, and the key step is to establish an analogous averaging principle on shorter time scales. As a consequence of our main theorem, we obtain a homogenization result for the associated advection-diffusion equation. We show that on intermediate time scales the effective equation is a fractional time PDE that arises in modelling anomalous diffusion.

10:00-10:50, J. Małecki (Wrocław Univ. Of Science and Techn.)   "Symmetric polynomials and stochastic particle systems".

Abstract: In the talk I will discuss the stochastic description of particle systems with repulsive forces, which naturally appear as eigenvalues of solutions of matrix-valued SDEs or systems of indeopendent one-dimentional processes conditioned to be non-colliding. I will present result related to existence, uniqueness and properties of such systems based on techniques related to basic symmetric polynomials. These are joint works with P. Graczyk, J-L. Perez and E. Mayerhofer. 

11:00-11:50, Jean-Philippe Anker (University of Orleans)  Dispersive PDEs on hyperbolic spaces

Abstract: The Schrödinger and the wave equations have better dispersive properties in negative curvature. We will illustrate this statement by considering a family of dispersive PDEs,
including the Schrödinger and the half wave equations, in two related settings :
hyperbolic spaces in the continous case and homogeneous trees in the discrete case.
Our results were obtained in collaboration with Vittoria Pierfelice, Yannick Sire and Maria Vallarino.

14:00-14:50, V. Kolokoltsov (Warwick University),  Sensitivity and regularity for the solutions to the McKean-Vlasov-type PDEs and SPDEs, with possibly singular coefficients, and applications.

Abstract: we shall discuss the well-posedness, regularity and sensitivity (smooth dependence on initial data)
for various classes of nonlinear Markov evolutions, deterministic and stochastic, arising from diffusions and/or stable-like processes. Possible applications include the interacting particle systems of statistical mechanics and mean-field games.

15:00-15:50, Lu Xu (Univ. Paris-Dauphine-CEREMADE)  Equilibrium fluctuation for a chain of anharmonic oscillators conserving momentum 

Abstract:  We consider a one-dimensional infinite chain of oscillators with noises. The interactions between oscillators are anharmonic and the noises conserve the total volume, momentum and energy  simultaneously. We investigate the fluctuation of the conserved quantities for equilibrium dynamics,  and obtain a linearized Euler system under hyperbolic space-time scale. 
This is based on a joint work in progress with Stefano Olla (Ceremade, Paris University Dauphine).

 

Tuesday, 5.06.2018 

9:00-9:50, R. Latała (Warsaw Univ.) Weak and strong moments of random vectors.

10:00-10:50, S. Olla (Univ. Paris-Dauphine-CEREMADE) Hydrodynamic limit in hyperbolic scaling with boundary conditions.

Abstract:  We study the hydrodynamic limit for a one dimensional isothermal anharmonic finite chain in Lagrangian coordinates with hyperbolic space-time scaling. The temperature is kept constant by putting the chain in contact with a heat bath, realised via astochastic momentum-preserving noise added to the dynamics. The noise is designed to be large at the microscopic level, but vanishing in the macroscopic scale. Boundary conditions are also considered: one end of the  chain is kept fixed, while a time-varying tension is applied to the other end.  We show that the microscopic volume stretch and the momentum converge (in an appropriate sense) to a weak solution of a system of hyperbolic conservation laws (isothermal Euler equations in Lagrangian coordinates) with boundary conditions.  This result includes the shock regime of the system. Work in collaboration with Stefano Marchesani (GSSI, L’Aquila).

11:00-11:50, K. Bogdan (Wrocław Univ. Of Science and Techn.) Fractional Laplacian with Hardy potential.

14:00-14:50, A. Kulik (Ukrainian Academy of Science) Generalized couplings and ergodic rates for SPDEs.

Abstract: By means of a new  generalized coupling method, we develop a general algorithm for proving ergodic rates for Markov processes that may lack the strong Feller property. This algorithm is applied to show exponential ergodicity of a variety of nonlinear stochastic partial differential equations with additive forcing, including the 2D stochastic Navier--Stokes equation. The talk is based on the joint research with O.Butkovsky and M. Scheutzow (Technical University, Berlin).  

15:00-15:50, K. Oleszkiewicz  (Warsaw Univ.) Strong contraction of Markov semigroups. 

Abstract: The talk will be based on a joint paper with Steven Heilman and Elchanan Mossel. We will discuss L^p-contractive properties of Markov semigroups implied by the spectral gap inequality.

19:00 Restaurant Dinner

 

 

Wednesday, 6.06.2018,

9:00-9:50,K. Szczypkowski (Wrocław Univ. Of Science and Techn.) 

10:00-10:50, Lihu Xu, Univ. of Macau, Malliavin-Stein's approach to multi-variate approximation in Wasserstein distance

Abstract: Stein's method has been widely used for probability approximations. However, in the multi-dimensional setting, most of the results are for multivariate normal approximation or for test functions with bounded second- or higher-order derivatives. For a class of multivariate limiting distributions, we use Bismut's formula in Malliavin calculus to control the derivatives of the Stein equation solutions by the first derivative of the test function. Combined with Stein's exchangeable pair approach, we obtain a general theorem for multivariate approximations with near optimal error bounds on the Wasserstein distance.We apply the theorem to the unadjusted Langevin algorithm. This is joint work with Qi-Man Shao and Xiao Fang.