Abstracts

Radosław Adamczak Polish Academy of Sciences and University of Warsaw Concentration of measure for non-Lipschitz functions I will review concentration results for non-Lipschitz functions obtained in recent years with various coauthors, including inequalities for polynomials in independent random variables, higher order concentration inequalities, and (time permitting) self normalized moment estimates for convex functions.   David Applebaum Univ. of Sheffield Invariant Feller Processes on Compact Symmetric Spaces We investigate some properties of invariant Feller processes on compact symmetric spaces $M$. All such spaces may be realised as quotients of compact Lie groups $G$ by closed subgroups $K$. The invariance of the title refers to $G$-invariance of the transition probabilties of the process in question. It turns out that all such processes are projections of Levy processes in $G$, and this means that a lot of global Fourier analytic tools become available. We investigate (i) The conditions for existence of a continuous transition density, and representation of this density in terms of spherical functions. (ii) Representation of the semigroup and generator as pseudo--differential operators. If time allows, we will discuss going beyond "invariant" Feller processes, using a new global characterisation of operators satisfying the positive maximum principle on a Lie group. This is based on joint work with Trang Le Ngan.

 

Zdzisław Brzeźniak

York Univ., UK

Fractionally dissipative stochastic quasi-geostrophic type  equations
on $R^d$.

Stochastic fractionally dissipative quasi-geostrophic type equation on ${\mathbb{R}}^{d}$ with a multiplicative Gaussian noise is considered. We prove the existence of a martingale solution. The construction of the solution is based on the classical Faedo-Galerkin approximation, the compactness method, and the Jakubowski version of the Skorokhod Theorem for nonmetric spaces. In the 2D sub-critical case  we prove also the pathwise uniqueness of the solutions. Based on a joint work with E. Motyl (Lodz).

 

Tymoteusz Chojecki

UMCS Lublin

Limit theorems for motions in random fields

We consider the trajectory of a tracer that is the solution of an ordinary differential equation X'(t) = V(t,X(t)), X(0) = 0, with the right hand side, that is a stationary, zero-mean, Gaussian vector field with incompressible realizations. It is known,  that X(t)/\sqrt(t) converges in law, as t → +∞, to a normal, zero mean vector, provided that the field V (t, x) is Markovian and has the spectral gap property. We extend this result to the case when the field is not Markovian and its covariance matrix is given by a completely monotone Bernstein function.

In our second result we move away from the stationarity assumption. A natural generalization of this assumption is a concept of local stationarity. We study the asymptotic behavior of a tracer moving in a divergence-free, locally stationary vector field. 

This is based on joint work with Tomasz Komorowski (Polish Academy of Science, Warsaw).

 

F. Cipriani

Dipartimento di Matematica - Politecnico di Milano

Logarithmic Sobolev Inequalities derived from Quantum Statistical Mechanics                                             

It is shown how Logarithmic Sobolev Inequalities follow, in a concise way, from a sub-exponential spectral growth rate assumption, via two fundamental properties of the Helmholtz Free Energy and Relative Entropy of states. The setting in completely general and examples will range among compact Riemannian manifolds, topological compact groups, dual of discrete groups, noncommutative tori, compact quantum groups and Ideal Bose gas.

 

Andreas Eberle

Univ Bonn

A coupling approach to the kinetic Langevin equation

The (kinetic) Langevin equation is an SDE with degenerate noise that describes the motion of a particle in a force field subject to damping and random collisions. It is also closely related to Hamiltonian Monte Carlo methods. An important open question is, why in certain cases kinetic Langevin diffusions seem to approach
equilibrium faster than overdamped Langevin diffusions.

So far, convergence to equilibrium for kinetic Langevin diffusions has almost  exclusively been studied by analytic techniques. In this talk, I present a probabilistic approach that is based on a specific combination of reflection and synchronous coupling of two solutions of the Langevin equation. The approach yields rather precise bounds for convergence to equilibrium at the borderline between the overdamped and the underdamped regime, and it may help to shed some light on the open question mentioned above.  

 

M. Hairer,

Imperial College London
Quasilinear singular SPDEs

 

Ifan Johnston

University of Warwick

Upper and lower bounds for the Green's function of time-fractional evolution equations.

We investigate bounds for the Greens function of equations $D^\beta f = Lf$, where $D^\beta$ is the Caputo fractional derivative, and the operator $L$ is either a second order parabolic operator in divergence form, or a homogeneous pseudo-differential operator with constant coefficients. We do this by using an operator valued Mittag-Leffler function to rewrite the solution of the equation in terms of stable densities, and using known estimates for these densities and estimates for the Greens function of $L$. This is joint work with Vassili Kolokoltsov.

 

Ansgar Juengel

TU Wien
Discrete entropy methods: combining PDE analysis, stochastics, and numerics

Entropy methods are a flexible and robust technique to investigate the large-time asymptotics for diffusion equations. For instance, the Bakry-Emery method allows for the derivation of exponential equilibration rates, even for nonlinear equations. It also provides proofs of convex Sobolev inequalities with computable constants. The extension to discrete systems is a great challenge in numerical analysis (e.g. for structure-preserving schemes) and in the theory of stochastic processes (e.g. for continuous-time Markov chains). In this talk, some results for time-discrete and/or space-discrete equations and the decay properties of their solutions are reviewed.  The techniques include discrete gradients, discrete gradient flows, the Bochner-Bakry-Emery method, and systematic integration by parts.

Christian Olivera

UNICAMP

2D Navier-Stokes equation with cylindrical fractional Brownian noise

This talk is in colaboration with Benedetta Ferrario(University of Pavia) We consider the Navier-Stokes equation on the 2D torus, with a stochastic forcing term which is a cylindrical fractional Wiener noise of Hurst parameter H. We prove a local existence and uniqueness result when 7/16<H<1/2 and a global existence and uniqueness result when 1/2<H<1. This talk is in colaboration with Benedetta Ferrario(University of Pavia).

 

Stefano Olla,

Ceremade Univ. Paris-Dauphine

Quasi-static hydrodynamic limits and their large deviation.

To model microscopically the quasi-static transformations of thermodynamics, we consider the hydrodynamic limits of interacting particle systems in a macroscopic space-time scaling such that the macroscopic time scale is larger than the typical relaxation time to equilibrium. The conditions at the boundary change in this macroscopic scale, that means very slowly. Large deviations from this limit are also studied.

Works in collaboration with Anna de Masi (U. L'Aquila).

 

Michela Ottobre

Heriot-Watt University

Title: Beyond the Hoermander condition  

In 1968 Hormander introduced a sufficient condition to ensure hypoellipticity of second order partial differential operators. As is well known, this seminal work of Hoermander had deep repercussions both in the analysis of PDEs and in probability theory. In this talk we will first review the Hormander condition by an analytical, probabilistic and geometric perspective. We then present the UFG condition, which is weaker than the Hormander condition. Such a condition was introduced by Kusuoka and Strook in the eighties. In particular, Kusuoka and Strook  showed that it is still possible to build a solid PDE theory for diffusion semigroups even in absence of the Hormander condition. We will therefore come to explain the significance of the UFG condition , in geometric,   probabilistic and analytical terms, and present new results (the first of this type) on the geometry and long time behaviour of diffusion semigroups that do not satisfy the Hormander condition.

 

Greg Pavliotis

Imperial College London

Long time behaviour and phase transitions for the McKean-Vlasov equation

We study the long time behaviour and the number and structure of stationary solutions for the McKean-Vlasov equation, a nonlinear nonlocal Fokker-Planck type equation that describes the mean field limit of a system of weakly interacting diffusions. We consider two cases: the McKean-Vlasov equation in a multiscale confining potential with quadratic, Curie-Weiss, interaction (the so-called Dasai-Zwanzig model), and the McKean-Vlasov dynamics on the torus with periodic boundary conditions and with a localized interaction. Our main objectives are the study of convergence to a stationary state and the construction of the bifurcation diagram for the stationary problem. The application of our work to the study of models for opinion formation is also discussed.

 

Katarzyna Pietruska-Pałuba

University of Warsaw

Extension theorem for nonlocal operators

We are concerned with the probabilistic method of solving the Dirichlet problem in a domain D ⊂ Rd for a nonlocal (L ́evy) operator with external boundary data g. The weak solution is given by the harmonic extension of g (via the Poisson kernel). We establish a Hardy-Stein-type identity between a weighted Sobolev semi-norm of g and another weighted Sobolev semi-norm of the harmonic extension.

The talk is based on the paper:
Krzysztof Bogdan, Tomasz Grzywny, Katarzyna Pietruska-Pal uba, Artur Rutkowski, Extension theorem for nonlocal operators, arXiv:1710.05880

 

E. Priola

Univ. Torino

Gradient estimates for SDEs  without monotonicity type conditions

Abstract: We prove gradient estimates for  transition Markov  semigroups $(P_t)$ associated to SDEs driven by  multiplicative Brownian noise  having possibly unbounded $C^1$-coefficients, without  requiring any monotonicity type condition.
In particular,   first  derivatives of coefficients  can grow polynomially and even exponentially. We establish  pointwise  estimates  with  weights for  $D_x P_t\varphi$   of the form
$$
{\sqrt{t}}  \, |D_x P_t \varphi (x) | \le 
c \, (1+ |x|^k) \, \| \varphi\|_{\infty}, 
$$
 $t \in (0,1]$, $\varphi \in C_b (\R^d)$, $x \in \R^d.$
To prove the result we use two main tools. First, we consider  a  Feynman--Kac semigroup with  potential $V$ related to the growth of the coefficients   and of their derivatives     for which we can use a Bismut-Elworthy-Li type formula.  Second,  we introduce  a new regular approximation for the coefficients of the SDE. At the end of the paper we  provide an   example of SDE with additive noise and  drift $b$ having sublinear growth together with its  derivative such  that uniform estimates for $D_x P_t \varphi$ without weights   do not hold. This is a joint work with G. Da Prato (Pisa).

 

Max v. Renesse

Leipzig Univ.

Reversible Coalescing-Fragmentating Wasserstein Dynamics on the Real Line

We introduce a family of reversible fragmentating-coagulating processes of particles of varying size-scaled diffusivity with strictly local interaction on the real line. The construction is based on a new family of measures on the set of real increasing functions as reference measures for naturally associated Dirichlet forms. The processes are infinite dimensional versions of sticky reflecting dynamics on a simplicial complex with positive boundary measure in all subcomplexes. Among other things we identify the intrinsic metric leading to a Varadhan formula for the short time asymptotics with the Wasserstein metric for the associated measure valued diffusion. Moreover, this processes represents a weak solution to a Dean-Kawasaki type SPDE for super-cooled liquids.  

 

Markus Riedle

King's College London

Variational solutions of SPDEs driven by cylindrical Levy processes

In this talk we consider SPDEs in the variational approach driven by a cylindrical L{\'e}vy process. Here, a cylindrical L{\'e}vy process is understood in the classical framework of cylindrical random variables and cylindrical measures, and thus, it can be considered as a natural generalisation of cylindrical Brownian motions or Gaussian space-time white noises.

The first part of the talk is devoted to the case of cylindrical L{\'e}vy processes with finite second moments. Under this assumption, we present a straightforward approach to stochastic integration and derive the existence of a variational solution. The latter can be established by mainly following  some standard arguments. The second part of the talk discusses the extension of the existence result to the general case of cylindrical L{\'e}vy processes without finite second moments. Since cylindrical L{\'e}vy processes are generalised processes not attaining values in the underlying Banach space, one cannot directly exploit standard approaches, such as stopping time arguments or a L{\'e}vy-It{\^o} decomposition.

Some parts of this talk are based on a joint work with Tomasz Kosmala.

 

C. Roberto

Univ. de Paris Ouest Nanterre la Defence

log Hessian estimates and the Talagrand Conjecture.

Motivated by Talagrand's conjecture about the regularization effect of the Ornstein-Uhlenbeck semi-group, we investigate lower bounds on the log Hessian of a family of diffusion semi-group (essentially perturbation of the Ornstein-Uhlenbeck semi-group) and prove that the conjecture holds. On the other side, we will also investigate similar questions for the (discrete) M/M/infinity queuing process on the integers and prove that, in that case, the conjecture is only partially true. Joint work with N. Gozlan, X.-M Li, M. Madiman and P.-M. Samson.

 

Francesco  Russo

ENSTA ParisTech,

Martingale problems, BSDEs, associated deterministic equations and applications to finance: the Markovian and the path-dependent framework

The aim of this talk consists in introducing a new formalism for the deterministic analysis  associated with backward stochastic differential equations  driven by general c\`adl\`ag martingales, coupled with a forward process.

When the martingale is a standard Brownian motion, and the forward process is a diffusion, the natural deterministic analysis is provided by the solution $u$ of a semilinear PDE of parabolic type coupled with a function $v$ which is associated with the  $\nabla u$, when $u$ is of class $C^1$ in space. When $u$ is only a viscosity solution of the PDE, the link associating $v$ to $u$ is not completely clear: sometimes in the literature it is called the {\it identification} problem. 

The idea is to introduce a suitable analysis to investigate the equivalent of the identification problem in a general Markovian (and non-Markovian) setting with a class of examples.
An interesting application concerns  the hedging problem under basis risk of a contingent claim $g(X_T,S_T)$, where $S$ (resp. $X$) is an underlying price of a traded (resp. non-traded but observable) asset, via the celebrated F\"ollmer-Schweizer decomposition. We revisit the case when the couple of price processes 
$(X,S)$ is a diffusion and we provide explicit expressions when $(X,S)$ is an exponential of additive processes.

The talk will be based on partial joint work with  Adrien Barrasso (ENSTA ParisTech) and Ismail Laachir (ZELIADE) 

 

Lenya Ryzhik

Stanford Univ.

The random heat equation in dimensions three and higher.

We consider  the heat equation with a multiplicative Gaussian potential in dimensions three and higher. We show that the renormalized solution converges to the solution of a deterministic diffusion equation with an effective diffusivity. We also prove that the
renormalized large scale random fluctuations are described by the Edwards-Wilkinson model, that is, the stochastic heat equation (SHE) with an additive white noise, with an effective diffusion and an effective variance. This is a joint work with Alex Dunlap, Yu Gu and Ofer Zeitouni. 

 

Marielle Simon

INRIA, Lille
Hydrodynamic limits for chains of oscillators and Wigner distributions.

In a collaborative work with T. Komorowski and S. Olla, we study the macroscopic behavior of a chain of N coupled harmonic oscillators. In order to provide the system with good ergodic properties, we perturb the Hamiltonian dynamics with random flips of velocities, in such a way as to conserve the energy of particles, and such that momentum conservation is no longer valid. We prove that in a diffusive space-time scaling limit the profiles corresponding to the two conserved quantities converge to the solution of a diffusive system of differential equations. While the elongation follows a simple autonomous linear diffusive equation, the evolution of the energy depends on the gradient of the square of the elongation. 

We follow an approach based on Wigner distributions, which permit to control the energy distribution over various frequency modes and provide a naturalseparation between mechanical and thermal energies. In the macroscopic limit we prove that locally the thermal energy spectrum has a constant density equal to the local thermal energy (or temperature), i.e. that the system is, at macroscopic positive times, in local equilibrium, even though it is not at initial time.

 

Tomasz Szarek

Univ. Of Gdansk

Random iterations of homeomorphisms on the circle

This talk is devoted to the problem of ergodicity for iterated function systems consisting of homeomorphism defined on the circle. 
We will provide also the rate of convergence and obtain the Central Limit Theorem and Law of the Iterated Logarithm as its consequence.
This is a joint work with A. Zdunik

 

Feng-Yu Wang

Swansea University

Gradient Estimates on Dirichlet and Neumann Eigenfunctions

By using stochastic analysis on Riemannian manifolds, two-sided gradient estimates are derived for higher order Dirichlet and Neumann eigenfunctions. Boundary gradient estimates are also discussed.

 

Ewelina Zatorska

Univ. College, London

On the hydrodynamic models with non-local forces: Critical thresholds and large-time behavior.

I will discuss the one-dimensional pressureless Euler-Poisson equations with a linear damping and non-local interaction forces. These equations are relevant for modelling  collective behavior in mathematical biology. We provide a sharp threshold between the supercritical region with finite-time breakdown and the subcritical region with global-in-time existence of the classical solution. We derive an explicit form of solution in Lagrangian coordinates which enables us to study the time-asymptotic behavior of classical solutions with the initial data in the subcritical region. 
I will also mention more recent result for the multi-dimensional Navier-Stokes type system with similar nonlocal forces and a local repulsion modelled by the pressure. 

 

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