CNRSPAN Mathematics Summer Institute, Cracow
28 June  4 July, 2015
In collaboration with the Institute of Mathematics, Jagiellonian
University, Institute of Mathematics, Polish Academy of Sciences, Faculty
of Applied Mathematics, AGH University of Science and Technology,
CNRS, Imperial College London, IMP Universite Paul Sabatier Toulouse, and
ANR project STAB.
Organizers: Dominique Bakry (Toulouse), Szymon Peszat (Cracow), and
Boguslaw Zegarlinski (London).
The meeting will review recent results in the area of
analysis/stochastics. Besides a number of presentation by international
participants, the meeting will include two minicourses (suitable
for PhD students and young researchers).
The venue of the workshop is:
Room 1016, the main building of the Faculty of Mathematics and Computer
Sciences, Jagiellonian University, Lojasiewicza 6, 30348 Krakow.
Unfortunatelly the building is not in the center of the town. You may
need about 40 minutes to go there. In order to help you to reach the
place, the first day of the Workshop, Mondey 28.06 at 9 am there will be some young local participants
waiting in the receptions of Hotel Classic on Tomasza Street and
Cybulskiego GuestRooms (Cybulskiego Street) where most of the participants have their
accomodation.
Previous and related activities:

Workshop on Stochastic Analysis and Related Fields, Cracow 20  22 July,
2009

Workshop on Analysis and Stochastic Analysis, Cracow 11  15 July, 2011

CNRSPAN Mathematics Summer Institute, Cracow 9  13 July, 2012

CNRSPAN Mathematics Summer Institute, Cracow 1  7 July, 2013

Noncommutative Workshop, Cracow 912 September 2013

CNRSPAN Mathematics Summer Institute, Cracow 26 June  4 July, 2014

Noncommutative Workshop, Cracow 2025 September 2015
Minicourses:
 Eric Carlen: Functional inequalities and evolution equations
Abstract: Functional inequalities are an essential tool for understanding
the behavior of solutions of PDE's and other sorts of evolution equations.
At the same time, monotonicity along various evolution processes can often
be used to prove functional inequalities. This course will
focus on some recent examples in which this interplay between evolution
processes and functional inequalities has been fruitful, and it will also
introduce some open problems in the field. There will a special emphasis
on stability results for sharp inequalities.
 Ansgar Juengel, Stochastic PDEs, Coercive inequalities and
numerical modeling
Abstract: Entropydissipation methods have been developed recently to
investigate the qualitative behavior of solutions to nonlinear partial
differential equations and to derive explicit or optimal constants in
convex Sobolev inequalities. It turned out that these methods also help
for the global existence analysis and the design of stable numerical
schemes. In this lecture series, we will highlight some of the aspects of
entropy methods, in particular for linear and nonlinear FokkerPlanck
equations, crossdiffusion systems from biology, and MaxwellStefan
systems for multicomponent fluid mixtures. New analytical tools are
the boundednessbyentropy principle and systematic integration by
parts. Connections to Markov processes will be highlighted too.
List of participants:
Narcisa Apreutesei (Iasi), Dominique Bakry (Toulouse), Witold
Bednorz (Warsaw), Leokadia
BiałasCież (Kraków), Dorota
Bors (Łódz), Zdzisław Brzeźniak (York), Jarosław
Duda (Kraków), Eric Carlen, (Rutgers), Gabriela Ciołek
(Kraków), Aneta
Gospodarczyk (Gdańsk), Jacek Gułgowski (Gdańsk), Franca
Hoffman (London), Ansgar Juengel (Vienna),
Piotr Knosalla (Opole), Tomasz Kosmala (Kraków), Anna Kozhina
(Moscow), Songzi Li (Toulouse), Mateusz Majka (Kraków, Bonn), Ewa Marciniak (Kraków), Laurent Miclo (Toulouse), Pierre
Monmarché (Toulouse), Elżbieta Motyl (Łódz), Katarzyna
PietruskaPałuba
(Warsaw), Szymon Peszat
(Kraków), Przemysław Rola (Kraków), Robert Stańczy
(Wrocław), Anna Sulima
(Kraków), Tomasz Szarek
(Gdańsk), Hanna Wojewódka (Gdańsk), Andrei Velicu
(London), Frantisek Zak (London), Dariusz Zawisza (Kraków), Boguslaw Zegarlinski (London), Maria Ziemianska (Leiden),
Willem van Zuijlen (Leiden), Andrzej Zuk (Paris), Wen Yue (Vienna).
For more information contact S. Peszat, email napeszat[AT]cyfkr.edu.pl
Talks (not complete):
 Narcisa Apreutesei: ReactionDiffusion Waves with Nonlinear
Boundary Conditions in Connection with Inflammatory Diseases
Abstract: We are concerned with the existence of travelling wave solutions
for reactiondiffusion equations with nonlinear boundary conditions. Such
equations have been proposed to model the development of atherosclerosis
or other inflammatory diseases. Monostable and bistable equations are
approached. In the bistable case one employs the LeraySchauder method,
that is based on a topological degree for elliptic problems in unbounded
domains and on a priori estimates of solutions. Fredholm property for
associated linearized operators is proved and properness of the semilinear
operators is studied in weighted Holder spaces.
 Witold Bednorz: The generic chaining approach to the regularity
of paths
Abstract: In my talk I will show how to use the method of chaining to get
results on the regularity of paths of processes which increments can be
controlled. I will mention applications to the analysis of log concave
vectors as well the study of processes of truncated variation on the real
line. I should also say something on the concentration type inequality for
compressed sensing in the setting of independent entries of exponential
decay.
 Zdzisław Brzeźniak: Large deviations principle for
invariant
measures for the 2D stochastic NavierStokes Equations
Abstract: Based on a recent work with S Cerrai and M Freidlin on the
quasipotential for the 2D stochastic NavierStokes Equations (to
appear in PTRF 2015) we prove the Large deviations principle for
invariant measures for these equations driven by an additive nuclear
gaussian noise and we identify the action functional. This talk is
based on a joint work with S Cerrai.
 Jarosław Duda: Maximal Entropy Random
Walk in stochastic models, information theory and network analysis
Abstract: The standard way of choosing stochastic models (transition
probabilities) in many cases turns out to be in disagreement with
experiment, correctly described by quantum mechanics. For example it would
allow electrons to freely travel through defected lattice of
semiconductor, while we know that it is not a conductor  these electrons
are statistically prisoned (Anderson localization). Maximal Entropy Random
Walk (MERW) allows to understand and repair this disagreement by choosing
the transition probabilities accordingly to the basic for statistical
physics: maximum entropy principle (Jaynes). MERW leads to stationary
probability distribution exactly like predicted by quantum mechanics, for
example with electrons prisoned in entropic well of semiconductor. I will
also tell about other applications of MERW: to maximize capacity of
informational channel and to analyze complex networks/graphs.

Franca Hoffmann: Asymptotic behaviour of diffusing and selfattracting
particles
Abstract: We study interacting particles behaving according to a
reactiondiffusion equation with nonlinear diffusion and nonlocal
attractive interaction. This class of equations has a very nice gradient
flow structure that allows us to make links to variations of wellknown
functional inequalities (HardyLittlewoodSobolev inequality, logarithmic
Sobolev inequality). Depending on the nonlinearity of the diffusion, the
choice of interaction potential and the dimensionality, we obtain
different regimes. Our goal is to understand better the asymptotic
behaviour of solutions in each of these regimes, starting with the
faircompetition regime where attractive and repulsive forces are in
balance. This is joint work with José A. Carrillo and Vincent Calvez.

Songzi Li: Matrix Dirichlet
processes.
 Mateusz Majka: Couplings and contractions for Lévydriven
stochastic differential equations
Abstract: We present an idea for a coupling of solutions of stochastic
differential equations driven by Lévy processes. Then we use this
coupling to obtain exponential contractivity for the semigroups
generated by those equations in a specially constructed Kantorovich
distance. As a consequence, we obtain upper bounds for those
semigroups in the total variation and L^1Wasserstein distances.
 Pierre
Monmarché: Hypocoercive simulated annealing in metastable
settings.

Elżbieta Motyl: Invariant measures for stochastic NavierStokes
equationsa in unbounded domains
Abstract: Using the MaslowskiSeidler theorem and bwFeller property we
prove the existence of an invariant measure for stochastic NavierStokes
equations with multiplicative noise in 2D possibly unbounded Poincaré
domains. The talk is based on joint work with Z. Brzeźniak and M.
Ondrejat.
 Katarzyna PietruskaPałuba: Semigroups of Lévy processes
with
Poissonian interaction: annealed and quenched asymptotics
Abstract: For the semigroup of a Lévy process evolving in a random
Poissonian
environment in R^d, we investigate its longtime behaviour.
We consider two types of asymptotics: after averaging with respect to
the random medium (the socalled annealed asymptotics), and also
almostsure with respect to the random medium (the socalled quenched
asymptotics). For Brownian motion, such problems were addressed by
Donsker and Varadhan (annealed), Sznitman (quenched). In that case,
the annealed asymptotics and the quenched asymptotics are different.
For Lévy processes, the two rates may differ, but they can coincide
as well, which is a new phenomenon.
We will discuss precise rate functions, both annealed and quenched, for
particular processes (e.g. stable, relativistic).
The results of this talk were obtained jointly with Kamil Kaleta.

Robert Stańczy: Entropy Methods for Systems of Gravitating
Particles
Abstract: We provide a few results on applications of the entropy methods
to the
systems of diffusing and gravitating particles, modeled by the generalized
SmoluchowskiPoisson equation. We consider both the case of fixed
temperature or the energy of the system. Some results concern the systems
of particles obeying different statistics i.e. the MaxwellBoltzmann,
FermiDirac, BoseEinstein ensembles or the one modeling polytropic stars.
A common feature of all the mentioned models is the pressure in the
selfsimilar form. Parts of the results were obtained together with Piotr
Biler and Jean Dolbeault.

Tomasz Szarek: Stability of iterated function system on the circle
Abstract: We prove that any Iterated Function System of circle
homeomorphisms such
that one of them possesses a dense orbit has a unique invariant measure
and is asymptot
ically stabe. The Strong Law of Large Numbers (SLLN) for trajectories
starting from an
arbitrary point for such function systems is also proved. (joint work with
A. Zdunik).
 Hanna Wojewódka: Ergodic properties of iterated function
systems
generated by some Markov operators
Abstract: The talk summarizes the research focused on examining ergodic
properties of stochastic dynamical systems generated by some Markov
operators defined on Polish spaces. The mathematical model, which is
analyzed, describes the process of cell division. First cell cycle models,
which were in fact the inspiration for the research, were described by
J.J. Tyson and K.B. Hannsgen (1988), as well as by A. Lasota and M.C.
Mackey (1999).
The ergodic description of the generalised cell cycle model is the aim of
the presentation. The main result, i.e. the estimation of the rate of
convergence of Markov operators to the unique invariant measure, was
obtained thanks to the techniques based on the construction of coupling
measure on the whole trajectories, adapted from Hairer (2002). Proving
that the rate is exponential allowed for further investigation of ergodic
properties such as the Central Limit Theorem or the Law of the Iterated
Logarithm. The results presented during the talk may be interesting not
only from mathematical but also from biological perspective.

Frantisek Zak:
Convergence to the equilibrium of infinite system of interacting
diffusions
Abstract: We outline new method, how to construct and prove pointwise
ergodicity properties of infinite system of interacting diffusions. Our
approach can cover nontrivial cases, where the generator of diffusion is
degenerate elliptic, such as Heisenberg group.
 Dariusz Zawisza:
HJB equations for ergodic discounted stochastic control with unbounded
discount rate
Abstract: We consider the semilinear HJB equation for an infinite horizon
stochastic control problem with the stochastic discount rate which might
be an unbounded function of the control process. We provide a set of
general assumptions to ensure that there exist a smooth classical
solution to that equation. The proof is based on the approximation of the
infinite horizon model using finite horizon control problems.

Andrzej Zuk: Random walks on random symmetric groups.
Tentative Program:
Sunday 28 June
Arrival Day
Monday 29 June
10.00  11.00 Witold Bednorz
11.00  11.15 T&C
11.15  12.15 Andrzej Zuk
14.00  14.45 Jarosław Duda
14.45  15.00 T&C
15.00  15.30 Dariusz Zawisza
15.35  16.25 Frantisek Zak
Tuesday 30 July
10.00  11.30 Ansgar Juengel (Minicourse 1)
11.30  11.45 T&C
11.45  12.45 Narcisa Apreutesei
14.00  15.30 Eric Carlen (Minicourse 1)
15.30  15.45 T&C
15.45  16.30 Katarzyna PietruskaPałuba
Wednesday 1 July
10.00  11.30 Eric Carlen (Minicourse 2)
11.30  11.45 T&C
11.4513.15 Ansgar Juengel (Minicourse 2)
14.00  15.30 Eric Carlen (Minicourse 3)
15.30  20.30 Excursion
Thursday 2 July
10.00  11.300 Ansgar Juengel (Minicourse 3)
11.30  11.45 T&C
11.45  12.45 Tomasz Szarek
14.00  14.45 Mateusz Majka
14.50  15.30: Franca Hoffmann
15.30  15.45 T&C
15.45  16.30 Robert Stańczy
18.30  20.30 History of Science Evening, Aula Jagiellońska, Collegium
Maius.
Professor Michał Kokowski (PAN) will give a lecture
"Nicolaus Copernicus (1473–1543).
His curriculum vitae,
mathematical
readings,
two scientific (r)evolutions, and… hindex"
Friday 3 July
10.00  11.00 Zdzisław Brzeźniak
11.00  11.15 T&C
11.15  12.00 Pierre Monmarché
12.05  12.35 Songzi Li
13.30  14.30 Elżbieta Motyl
14.30  14.40 T&C
14.40  15.25 Hanna Wojewódka
Saturday 4 July
10.00  14.00 Discussion Groups
15.00  17.00 Discussion Groups
Sunday 5 July
Departure Day