PDEs/SPDEs & Functional Inequalities

01.04.2018 - 30.06.2018 | Będlewo & Warsaw


10th of April

Room 106 at 11:00

Z. Brzeźniak

York Univ.

Title "Quasipotential for Stochastic Navier-Stokes and Landau-Lifshitz-Gilbert Equations".
"Quasipotential for Stochastic Navier-Stokes and Landau-Lifshitz-Gilbert Equations".

Abstract.  The notion of quasipotential is useful in the study of small noise asymptotics, e.g. large deviations and/or rare     events, of solutions and invariant measures of non-potential Stochastic Partial Differential Equations, as for instance. 
    Stochastic Navier-Stokes and Landau-Lifshitz-Gilbert Equations. In some cases one can find an explicit formula for the     quasipotential. This talk is stems from various joint works of the speaker with S. Cerrai, M. Freidlin, B. Goldys, T. Jegaraj, L. Li and E. Hausenblas. 


Room 106 at 12:45

Bogusław Zegarliński

Imperial College London

Title “Challenges in coercive inequalities" 

Abstract. I will talk about recent progress in understanding log-Sobolev inequality in a challenging commutative and noncommutative setup, including analysis on nipotent Lie groups and noncommutative algebras. 


17th of April

Room 106 at 11:30

Stefano Olla,

CEREMADE, Univ. Paris-Dauphine, Paris 

Title: Hydrodynamic limit for a harmonic chain with random masses

Abstract: We consider the hamiltonian dynamics of an unpinned chain of harmonic oscillators with random masses. 
Under a hyperbolic rescaling of space and time we obtain the full set of Euler equations for the macroscopic evolution of energy, momentum and volume stretch. This is in contrast with the deterministic masses case, where there is no autonomous macroscopic evolution for the energy field if the system is not in thermal equilibrium. 
The dynamics is completely integrable, but Anderson localization, due to the randomness of the masses, freeze the evolution of the thermal (microscopic) frequency modes, allowing a clear separation of scales from the macroscopic modes that are governed by the Euler equations. 
Work in collaboration with Cedric Bernardin (U. Nice) and Francois Huveneers (CEREMADE).


Room 106 at 13:15,

Michela Ottobre,

Heriot-Watt University

Title: A one-dimensional model for self-propelled diffusions

Abstract.  One of the new challenges of statistical mechanics arises from the study of interacting particle systems of self-propelled particles. Such models are at the root of many biological phenomena, such as bacterial migration, flocking of birds etc.  In this talk we will consider a non-linear PDE for a Viksek-type model. The PDE at hand is  i) not in gradient form and ii) the linear part  of the equation is non-uniformly elliptic  (but hypoelliptic instead). Moreover, as typical in this framework, the dynamics exhibits multiple equilibria (stationary states), corresponding to formation of different coordinated motions.  This is a joint work with P.Butta (La Sapienza, Rome), F. Flandoli (Scuola Normale, Pisa) and B. Zegarlinski (Imperial College).


20th of April

Room 106 at 10:30-12:00

Xue-Mei Li
Imperial College, London
Title: Heat equation with a potential

Consider an elliptic diffusion operator L and a real valued function V.
Let q_t be the fundamental solution to d/dt=(L-V). We discuss gradient and Hessian 
estimates for nabla log q_t. For V=0, this is used to understand Brownian bridges.


Room 403 at 14:15-15:45

Martin Hairer 

Imperial College, London

Title: A new universality class in 1+1 dimensions.


7th and 8th of May

Mini Course

Room 106 at 10:30

Z. Brzeźniak

York Univ.

Title: "Stochastic Landau-Lifshitz-Gilbert Equations driven by Levy processes"

Abstract: I will speak about the existence of weak martingale solutions stochastic Landau-Lifshitz-Gilbert Equations in the Marcus canonical form driven by pure jump Levy processes. I will also speak about the Quasipotential  Landau-Lifshitz-Gilbert Equations motivated by large deviations stochastic Landau-Lifshitz-Gilbert Equations driven by Wiener process.  



11th of May

Room 106 at 11:00-11:50

M. Maurelli

University of York

Title: Stochastic 2D Euler equations: well-posedness for bounded vorticity (joint work with Zdzislaw Brzezniak and Franco Flandoli)


Room 403 at 14:10-15:00

F. Russo


Title: Forward-Backward SDEs driven by a cadlag martingale: associated deterministic equations and applications.


14th and 16th of May

Mini Course

14.05.2018 at 10:30-12:00, Room 106

16.05.2018 at 10:30-12:00, Room 106

Enrico Priola

Dipartimento di Matematica Universita' di Torino

Title:Poisson stochastic process and basic parabolic Schauder and Sobolev  estimates  

Abstract: We show  how  knowing Schauder  or Sobolev-space estimates for the one-dimensional heat equation allows one to derive their multidimensional analogs for equations with oefficients depending only on time variable with the  same constants as in the case of the one-dimensional heat equation.
The method is   quite general and is  based on using the Poisson stochastic process.  It also applies to equations involving non-local operators.  It looks like no other method is available at this time and it is a very challenging problem
to find a purely analytic approach to proving such results. 
This is a joint work with N.V. Krylov


15th of May

Room 106 at 11:00-11:50

A. Guillin

Univ. Of Clermont-Ferrant

Title:Convergence to equilibrium for the kinetic Fokker-Planck equation


Room 106 at 13:30-14:20

François Bolley

Univ. Paris 6

Title: Coulomb gas in two dimensions


18th of May

Room 106 at 11:00-11:50

 J. A. Carrillo

Imperial College London

Title: Swarming models with local alignment effects: phase transition & hydrodynamics

Abstract: I will make a review of swarming models with repulsive-attractive effects focusing on two new aspects: phase transitions for the local Cucker-Smale type model and self-organized hydrodynamics of the Vicsek model with fixed speed from asymptotic speed Cucker-Smale models by penalization. In short, we will show that the asymptotic speed Cucker-Smale model behaves in terms of hydrodynamics and phase transitions as the Vicsek model with fixed speed for large friction parameter.


21th and 23th of May

Mini Course

21.05.2018 at 10:30-12:00, Room 106

23.05.2018 at 14:00-15:30, Room 403

I. Gentil

Title: When Otto meets Newton and Schrödinger, an heuristic point of view. 

Abstract: I’m going to explain the Schrödinger problem and his links with the optimal transport and the Newton equation with respect to the Otto calculus.


22th of May

Room 106 at 11:00-11:50

Dan Crisan

Imperial College

Title: "High order discretizations for the solution of the nonlinear filtering problem"

The solution of the continuous time filtering problem can be represented as a ratio of two expectations of certain functionals of the signal process that are parametrized by the observation path. We introduce a class of discretization schemes of these functionals of arbitrary order. The result generalizes the classical work of Picard, who introduced first order discretizations to the filtering functionals. For a given time interval partition, we construct discretization schemes with convergence rates that are proportional with the m-power of the mesh of the partition for arbitrary positive integer m. The result paves the way for constructing high order numerical approximation for the solution of the filtering problem. The talk is based on the paper

DC, Salvador Ortiz-Latorre, A high order time discretization of the solution of the non-linear filtering problem https://arxiv.org/abs/1711.08012 

Room 106 at 13:30-14:20

Andreas Eberle

Univ. Of Bonn

Sticky couplings for diffusion processes

Carefully constructed Markovian couplings and specifically designed Kantorovich metrics can be used to derive relatively precise and long-time stable bounds on the distance between the laws of two diffusions. In this talk, I will explain how to construct a new family of couplings that are sticky on a hyperplane. In the case of two overdamped Langevin diffusions with the same drift, these reduce to standard reflection couplings, but for processes with different drifts, the reflection coupling is replaced by a „sticky coupling“ where the distance between the two copies is bounded from above by a one-dimensional diffusion process with a sticky boundary at 0. This new type of coupling leads to long-time stable bounds on the total variation distance between the two laws. Similarly, two kinetic Langevin processes can be coupled using a particular combination of a reflection and a synchronous coupling that is sticky on a hyperplane. This can be applied to derive new bounds of kinetic order for convergence to equilibrium at the borderline between the overdamped and the underdamped regime. Related approaches are also useful when studying mean-field interacting particle systems, McKean-Vlasov diffusions, or diffusions on infinite dimensional state spaces.

(joint work with Arnaud Guillin and Raphael Zimmer). 


23th and 25th of May

Mini Course

23.05.2018 at 10:30-12:00, Room 106

25.05.2018 at 10:30-12:00, Room 106

B. Gołdys

Univ. Of Sydney

Parabolic PDEs with boundary noise

ABSTRACT. We will present recent developments in the analysis of second order parabolic PDEs driven by noise that enters the equation as a Dirichlet boundary condition.  The talks will be based on a joint work in progress with Szymon Peszat. If time permits, we will also present some results on equations with dynamic boundary conditions. 


28th and 30th of May

Mini Course

28.05.2018 at 10:30-12:00, Room 106

30.05.2018 at 10:30-12:00, Room 106

Prof. S. Olla

CEREMADE, Univ. Paris-Dauphine


This course is an introduction to Thermodynamics and Statistical Mechanics from the point of view of space-time scaling limit. The aim is to explain the main ideas in one concrete model, more than constructing a general theory.

Thermodynamic laws are defined and explained how can be deduced from the microscopic dynamics of molecules through space-time macroscopic limits. Thermodynamics is obtained as macroscopic theory, i.e., valid on macroscopic space-time scales, while statistical mechanics provides the microscopic model. This means that the objects of thermodynamics are those macroscopic complex systems that satisfy the thermodynamic principles, while statistical mechanics explains how this complexity arise from dynamics of systems with a very large number of components. The central point of this connection is the identification of the thermodynamic entropy, a function of the thermodynamic macroscopic equilibrium states, introduced by Clausius using Carnot cycles, with the Boltzmann definition of entropy in statistical mechanics, as logarithm of the number of microstates corresponding to the given macroscopic state. Boltzmann and Planck discovered this identification at the end of 19th century, but in order to understand it, we need to obtain, from microscopic dynamics, the quasi-static thermodynamic transformations constituting the Carnot cycle, isothermal and adiabatic.
This aspect differentiates this course from more classical courses in statistical physics, where only the equilibrium properties of the systems are studied.

Classical thermodynamics concerns transformations from an equilibrium state to another. I will also illustrate how some ideas generalize to transitions between non-equilibrium stationary states.


29th of May

Room 106 at 11:00-11:50

Ewelina Zatorska


Title: Finite-Energy Solutions for the Compressible Two-Fluid System 

I will discuss novel compactness tool for Navier-Stokes equations that we used in order to prove existence of global in time weak solutions to a compressible two-fluid Stokes system. We considered a single velocity field and algebraic closure for the pressure law. The constitutive relation involves  densities of both fluids through  an implicit function. I will explain the difference between classical Lions-Feireisl approach and the Bresch-Jabin approach developed for mono-fluid setting with non-monotone pressure law.
Next, I will construct weak solutions via unconventional approximation using  the Lagrangian formulation, truncations and stability  result of trajectories for rough velocity fields.

The talk is based on a joint result with D. Bresch and P.B. Mucha.

Room 106 at 13:30-14:20

A. Novikov

Penn. State University

Title: On random homogenization of  the G-equation


30th of May

Room 405 at 14:00-14:50

M. Majka

King’s College London

Title: Transportation inequalities via Malliavin calculus and coupling

Abstract: We introduce the topic of transportation cost-information inequalities that characterize concentration of probability measures. We focus on measures that are distributions of solutions to stochastic differential equations, with noise consisting both of a diffusion and a jump component. We show how to use the Malliavin calculus to obtain such inequalities. In particular, we combine the Malliavin calculus approach with the coupling technique, which allows us to obtain results under relatively weak assumptions on the drift in the equation.


7th and 8th of June

Mini Course

7.06. 2018 at 14:00-15:30, room 405

8.06.2018 at 10:30-12:00, room 106

Prof. Lihu Xu

Univeristy of Macau

Title: Approximation of stable law in Wasserstein distance by Stein's method

Abstract: We will first give a fast review of some preliminaries of stable law, stable processes, ergodicity of SDEs driven by stable noises, and then talk how to obtain the convergence rate of stable law in Wasserstein distance by Stein's method. If the time is permitted, we will give a sketch on using a method recently developed by Fang, Shao and Xu to sample high dimensional stable distribution by discretizing Ornstein-Uhlenbeck stable processes. This talk is based on the paper https://arxiv.org/abs/1709.00805


12th of June

Room 106 at 11:30-12:20

Prof. Jean-Philippe Anker

University of Orleans

Title : Sharp estimates for random walks on homogeneous trees and homogeneous buildings

Abstract : There are few examples where heat kernels or densities of random walks can be estimated everywhere, from above and from below, by the same plain expression, up to multiplicative constants. In this talk, which is based on a joint work with Bruno Schapira and Bartosz Trojan, I will consider the simple random walk first on homogeneous trees and next on affine buildings of type A2. The first case is elementary and will serve as an introduction to the second part, which I will try to make understandable for non specialists.


19th of June

Room 106 at 11:30-12:20

Prof. Ansgar Jüngel

Technische Univ. Vienna

Deterministic and stochastic cross-diffusion systems in population dynamics

Shigesada, Kawasaki, and Teramoto suggested in 1979 a reaction-diffusion system to describe the segregation of two population species. The system consists of strongly coupled diffusion equations with Lotka-Volterra reaction terms. Because of the lack of positive definiteness of the diffusion matrix, the mathematical analysis is delicate. Later, it was discovered that the system has a formal gradient-flow or entropy structure, allowing for Lyapunov functional type arguments which leads to the global existence of weak solutions. This structure even holds for an arbitrary but finite number of species under a detailed-balance condition. In this talk, we detail the entropy structure, the existence result,
and an extension to a stochastic version of the cross-diffusion system with multiplicative noise.

Room 106 at 14:00-14:50

Prof. E. Daus

Technische Univ. Vienna

Title: Reaction-cross diffusion systems and their derivation from microscopic models

Abstract: In the first part of my talk, I will present some recent results on the large-time asymptotics of weak solutions to Maxwell–Stefan diffusion systems for chemically reacting fluids with different molar masses and reversible reactions. The diffusion matrix of the system is generally neither symmetric nor positive definite, but the equations admit a formal gradient-flow structure which provides entropy (free energy) estimates. The main result is the exponential decay to the unique
equilibrium with a rate that is constructive up to a finite-dimensional inequality. The main difficulty comes from the fact that the reactions are represented by molar fractions while the conservation laws hold for the concentrations. The idea is to enlarge the space of $n$ partial concentrations by adding the total concentration, viewed as an independent variable, thus working with $n+1$ variables. This is a joint work with Ansgar Juengel and Bao Quoc Tang.
In the second half of the talk, I will review some recent results on the derivation of (reaction-) cross diffusion models from some underlying kinetic and particle models.


20th of June

Room 106 at 11:00-11:50

Prof. G. Corach

Instituto Argentino de Matemática "Alberto P. Calderón" CONICET


In this talk, I plan to discuss several operator inequalities, originally proved by Dixmier, Heinz, Cordes, Segal, among others, which describe some geometrical properties of the manifold of positive invertible operators of a Hilbert space.


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