CNRS-PAN Mathematics Summer Institute, Cracow 28 June - 4 July, 2015

• Workshop on Stochastic Analysis and Related Fields, Cracow 20 - 22 July, 2009
• Workshop on Analysis and Stochastic Analysis, Cracow 11 - 15 July, 2011
• CNRS-PAN Mathematics Summer Institute, Cracow 9 - 13 July, 2012
• CNRS-PAN Mathematics Summer Institute, Cracow 1 - 7 July, 2013
• Noncommutative Workshop, Cracow 9-12 September 2013
• CNRS-PAN Mathematics Summer Institute, Cracow 26 June - 4 July, 2014
• Noncommutative Workshop, Cracow 20-25 September 2015

• 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: Entropy-dissipation 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 Fokker-Planck equations, cross-diffusion systems from biology, and Maxwell-Stefan systems for multicomponent fluid mixtures. New analytical tools are the boundedness-by-entropy principle and systematic integration by parts. Connections to Markov processes will be highlighted too.

• Narcisa Apreutesei: Reaction-Diffusion Waves with Nonlinear Boundary Conditions in Connection with Inflammatory Diseases
  Abstract: We are concerned with the existence of travelling wave solutions for reaction-diffusion 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 Leray-Schauder 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 2-D stochastic Navier-Stokes Equations
  Abstract: Based on a recent work with S Cerrai and M Freidlin on the quasipotential for the 2-D stochastic Navier-Stokes 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 self-attracting particles
  Abstract: We study interacting particles behaving according to a reaction-diffusion equation with non-linear diffusion and non-local attractive interaction. This class of equations has a very nice gradient flow structure that allows us to make links to variations of well-known functional inequalities (Hardy-Littlewood-Sobolev inequality, logarithmic Sobolev inequality). Depending on the non-linearity 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 fair-competition 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 Lévy-driven 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^1-Wasserstein distances.
• Pierre Monmarché: Hypocoercive simulated annealing in metastable settings.
• Elżbieta Motyl: Invariant measures for stochastic Navier-Stokes equationsa in unbounded domains Abstract: Using the Maslowski-Seidler theorem and bw-Feller property we prove the existence of an invariant measure for stochastic Navier-Stokes equations with multiplicative noise in 2D possibly unbounded Poincaré domains. The talk is based on joint work with Z. Brzeźniak and M. Ondrejat.
• Katarzyna Pietruska-Pałuba: 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 long-time behaviour. We consider two types of asymptotics: after averaging with respect to the random medium (the so-called annealed asymptotics), and also almost-sure with respect to the random medium (the so-called 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 Smoluchowski-Poisson 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 Maxwell-Boltzmann, Fermi-Dirac, Bose-Einstein ensembles or the one modeling polytropic stars. A common feature of all the mentioned models is the pressure in the self-similar 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.