Probability and Analysis 2021

The conference Probability and Analysis 2021 was held online 24-28 May 2021. The previous meetings took place in Będlewo in 2012, 2015, 2017 and 2019 (see link). The conference covered selected topics in the intersections of probability and analysis. The meeting was an occasion to summarize current research, set future goals and launch joint projects. This time we had a less-packed online schedule to facilitate participation of academics from all over the world despite their busy daily routines. Our goal was to refresh old and find new connections and to keep the international collaboration in probability and analysis growing strong. We had 201 registered participants, including 69 students/PhD students. The registered participants were affiliated at 98 different universities.

Sessions

Each day we had a separate session of 3 lectures at 15:00, 16:00 and 17:00 (GMT + 2h), each lecture approx. 45 minutes + discussion/break. On Tuesday, Wednesday and Thursday there were also Contributed Sessions, which consisted of 2 lectures at 18:20 and 18:40 (GMT + 2h), each lecture approx. 15 minutes + discussion/break.

• 24May

     Branching and Markov additive processes

      Session Organizer: Victor Rivero, CIMAT

  
  
  15:00 - 16:00

  Loïc Chaumont, University of Angers

  Extinction times of multitype continuous-state branching processes.

  Abstract: We give necessary and sufficient conditions on the branching mechanism of a multitype continuous-state branching process for it to become extinct at a finite time. The probability of extinction is then given explicitely. This result extends Grey's condition known in the single type case. We also give conditions for the process to be conservative. This is a joint work with my PhD student Marine Marolleau.
  
  Abstract Video Slides
  
  

  16:00 - 17:00

  Leif Döring, University of Mannheim

  Results on stable SDEs (without drift)

  Abstract: One-dimensional Brownian SDEs have been studied for decades, many techniques were developed to give a full picture of existence/uniqueness/properties. The situation is different for SDEs driven by stable processes, only for $\alpha>1$ the classical local time tricks can be extended to prove Engelbert-Schmidt type results. We discuss recent results for $\alpha<1$.
  
  Abstract Video
  
  

  17:00 - 18:00

  José Luis Pérez-Garmendia, CIMAT

  The relative frequency between two continuous-state branching processes with immigration and their genealogy

  Abstract: When two (possibly different in distribution) continuous-state branching processes with immigration are present, we study the relative frequency of one of them when the total mass is forced to be constant at a dense set of times. This leads to an SDE whose unique strong solution will be the definition of a Lambda-asymmetric frequency process (Lambda-AFP). We prove that it is a Feller process and calculate a large population limit when the total mass tends to infinity. This allows us to study the fluctuations of the process around its deterministic limit. Furthermore, we find conditions for the Lambda-AFP to have a moment dual. The dual can be interpreted in terms of selection, (coordinated) mutation, pairwise branching (efficiency), coalescence, and a novel component that comes from the asymmetry between the reproduction mechanisms. In the particular case of a pair of equally distributed continuous-state branching processes, the associated Lambda-AFP will be the dual of a Lambda-coalescent. The map that sends each continuous-state branching process to its associated Lambda- coalescent (according to the former procedure) is a homeomorphism between metric spaces.
  
  Abstract Video Slides
• 25May

     Random structures

      Session Organizers: Dariusz Buraczewski, University of Wrocław, Wrocław
                                                    Nina Gantert, Technical University of Munich

  
  
  15:00 - 16:00

  Cecile Mailler, University of Bath

  Voronoi cells in random split trees

  Abstract: Take k nodes uniformly at random in a n-node graph, and let k competing epidemics spread along edges at constant speed from these initial nodes, infecting nodes when they reach them. A node can only get infected by one of the epidemics, the first one that reaches it. We are interested in the sizes of the final territories of the k epidemics, which we call the Voronoi cells of the k initial nodes. In this joint work with Markus Heydenreich (Munich) and Alex Drewitz (Cologne), we prove limiting theorems for the sizes of the Voronoi cells of k nodes taken uniformly at random in an n-node random split tree.
  
  Abstract Video
  
  

  16:00 - 17:00

  Kilian Raschel, Université de Tours

  Constructing discrete harmonic functions in wedges

  Abstract: I will present a systematic construction of signed harmonic functions for discrete Laplacian operators with Dirichlet conditions in the quarter plane. In particular, I will show that the set of harmonic functions is an algebra generated by a single element, which conjecturally corresponds to the unique positive harmonic function. This is a joint work (arXiv:2012.08947) with Viet Hung Hoang (University of Tours) and Pierre Tarrago (Sorbonne University).
  
  Abstract Video
  
  

  17:00 - 18:00

  Zakhar Kabluchko, Universität Münster

  Lah distribution, Stirling numbers, and random polytopes

  Abstract: We introduce a family of probability distributions defined in terms of Stirling and Lah numbers. On the one hand, these distributions appear in the formulas for the expected number of faces of a convex hull of a $d$-dimensional random walk. On the other hand, they admit a combinatorial interpretation in terms of random partitions and records. We shall derive various limit theorems for the Lah distributions and explain how these limit theorems imply phase transitions in the behavior of convex hulls of random walks as the dimension $d$ goes to infinity. This is joint work with Alexander Maryrych. Parts of the talk are based on a joint work with Vladislav Vysotsky and Dmitry Zaporozhets [https://arxiv.org/abs/1612.00249].
  
  Abstract Video Slides

   
              Contributed session

                  Session Chair: Artur Rutkowski, Wrocław University of Science and Technology

  
  

  18:20-18:40

  Grzegorz Serafin, Wrocław University of Science and Technology

  Dirichlet heat kernels in convex domains

  Abstract: Let $g_D(t,x,y)$ be the classical Dirichlet heat kernel on a given set $D\subset \mathbb R^n$. There is a great deal of results on estimates of such kernels, however, the vast majority of them share one common weakness: lower and upper bounds are not comparable, which is \mbox{a consequence} of different constants appearing in exponents. For example, let us recall the Zhang's result (2002) for $D$ being a bounded $\mathcal C^{1,1}$ domain or a complement of a closure of a bounded $\mathcal C^{1,1}$ domain: \begin{eqnarray*} c_1\left(\frac{\delta_D(x)\delta_D(y)}{t}\wedge 1\right)\frac{e^{-c_2|x-y|^2/t}}{t^{n/2}} \leq g_D(t,x,y) \leq c_3\left(\frac{\delta_D(x)\delta_D(y)}{t}\wedge 1\right)\frac{e^{-c_4|x-y|^2/t}}{t^{n/2}}, \end{eqnarray*} where $t<1$, $x,y\in D$ and $c_1,c_2,c_3,c_4$ are some positive constants. One can easily observe, by considering large values of $|x-y|^2/t$, that the ratio of the right-hand side to the left-hand one is unbounded. During this talk we will present general short-time estimates for the heat kernels on convex (not necessarily bounded) $\mathcal C^{1,1}$ domains that capture the proper exponential behavior. Furthermore, we will characterize two classes of domains for which the estimates are sharp, i.e. the upper and lower bounds coincide up to a multiplicative constant. Finally, the general results will be complemented with several examples.
  
  Abstract Video Slides
  
  

  18:40-19:00

  Julia Lenczewska, Wrocław University of Science and Technology

  Optimal Hardy inequality for the fractional Laplacian on $L^p$

  Abstract: We give an optimal Hardy inequality for a generalization of the quadratic form, the Sobolev-Bregman form, of the fractional Laplacian on $L^p(\mathbb{R}^d)$. We present an application of our results to the contractivity of the Feynman-Kac semigroup $\tilde{P}_t$ generated by $\Delta^{\alpha/2} + \kappa |x|^{-\alpha}$. The talk is based on a joint work with Krzysztof Bogdan, Tomasz Jakubowski and Katarzyna Pietruska-Pałuba [arXiv:2103.06550].
  
  Abstract Video Slides
• 26May

     Schrödinger operators on graphs and manifolds

      Session Organizers: Patricia Alonso Ruiz, Texas A&M University
                                                    Kamil Kaleta, Wrocław University of Science and Technology, Wrocław

  
  
  15:00 - 16:00

  Michael Hinz, Bielefeld University

  Self-adjoint Laplacians and symmetric diffusions on hyperbolic attractors

  Abstract: We consider attractors of hyperbolic dynamical systems, such as the dyadic solenoid or attractors of Plykin or Lorenz type. We use the 'natural tangential structure' provided by the stable manifold theorem and 'natural volume measures' of SRB- or Gibbs type to introduce self-adjoint Laplacians and symmetric diffusions via Dirichlet form theory. We discuss some properties and comment on the special case of geodesic flows. The presented results are joint with Shayan Alikhanloo (Bielefeld University).
  
  Abstract Video Slides
  
  

  Chair: Katarzyna Pietruska-Pałuba, University of Warsaw
  

  
  

  16:00 - 17:00

  Radosław Wojciechowski, York College (CUNY)

  Essential self-adjointness of the Laplacian on weighted graphs: stability and characterizations

  Abstract: We present some ideas concerning the essential self-adjointness of the Laplacian on weighted graphs. In particular, we discuss how properties of harmonic functions, stability and limit point-limit circle theory can be used to characterize essential self-adjointness of the Laplacian on birth-and-death chains. We then discuss some ideas on how to extend the result to Schrodinger operators. This talk is based on joint work with Atsushi Inoue and Jun Masamune.
  
  Abstract Video Slides
  
  

  Chair: Constanza Rojas-Molina, CY Cergy Paris Université
  

  
  

  17:00 - 18:00

  Matthias Keller, University of Potsdam

  From Hardy and Rellich inequalities to Agmon estimates on graphs

  Abstract: We present a method to obtain optimal Hardy weights on graphs. In particular, this yields the optimal constants and asymptotics for the Hardy weights on the Euclidean lattice. We proceed to discuss Rellich inequalities and Agmon estimates for (generalized) eigenfunctions. This includes joint work with Florian Fischer, Marius Lemm, Yehuda Pinchover and Felix Pogorzelski.
  
  Abstract Video Slides
  
  

  Chair: Luke Rogers, University of Connecticut
  

   
              Contributed session

                  Session Chair: Grzegorz Serafin, Wrocław University of Science and Technology

  
  

  18:20-18:40

  Wojciech Cygan, Wrocław University and TU Dresden

  Discrete-time Feynman-Kac operators and estimates for the corresponding harmonic functions

  Abstract: The classical Feynman-Kac formula can be studied in several different ways. From a probabilistic point of view, it can be seen as a representation of certain perturbations of the semigroup of operators related to Brownian motion. In the language of analysis, it can be viewed as a probabilistic representation for solutions to a class of parabolic partial differential equations. During the talk we present a discrete-time counterpart of the Feynman-Kac semigroup which is associated with a certian class of Markov chains defined on a weighted graph. These Markov chains satisfy the so-called direct step property and they include quite a few examples of big-step random walks on various graphs, such as integer lattice, Sierpinksi gasket, etc. We study the decay of functions that are harmonic with respect to such Feynman-Kac operators in infinite subsets of the graph and we briefly discuss some direct applications. This is a joint work with K. Kaleta and M. Śliwiński (Wrocław University of Science and Technology).
  
  Abstract Video
  
  

  18:40-19:00

  Michał Kijaczko, Wrocław University of Science and Technology

  On density of compactly supported smooth functions in fractional Sobolev spaces

  Abstract: We describe some sufficient conditions, under which smooth and compactly supported functions are or are not dense in the fractional Sobolev space $W^{s,p}(\Omega)$ for an open, bounded set $\Omega\subset\mathbb{R}^{d}$. The density property is closely related to the lower and upper Assouad codimension of the boundary of $\Omega$. We also describe explicitly the closure of $C_{c}^{\infty}(\Omega)$ in $W^{s,p}(\Omega)$ under some mild assumptions about the geometry of $\Omega$. The talk is based on a joint work with Bartłomiej Dyda [arXiv:2104.08953].
  
  Abstract Video Slides
• 27May

     High dimensional probability

      Session Organizers: Radosław Adamczak, University of Warsaw, Warsaw
                                                    Djalil Chafai, Université Paris-Dauphine

  
  
  15:00 - 16:00

  Charles Bordenave, Aix-Marseille Université

  Strong asymptotic freeness for independent uniform variables on compact groups

  Abstract: Asymptotic freeness of independent Haar distributed unitary matrices was discovered by Voiculescu. Many refinements have been obtained, including strong asymptotic freeness of random unitaries and strong asymptotic freeness of random permutations acting on the orthogonal of the Perron-Frobenius eigenvector. In this talk, we consider a new matrix unitary model appearing naturally from representation theory of compact groups. We fix a non-trivial signature, i.e. two finite sequences of non-increasing natural numbers, and for n large enough, consider the irreducible representation of Un associated to this signature. We show that strong asymptotic freeness holds in this generalized context when drawing independent copies of the Haar measure. We also obtain the orthogonal variant of this result. This is a joint work with Benoit Collins.
  
  Abstract Video Slides
  
  

  16:00 - 17:00

  Ronen Eldan, Weizmann Institute of Science

  Second-order bounds for quantitative FKG inequalities

  Abstract: Harris's correlation inequality states that any two monotone functions on the Boolean hypercube are positively correlated (this is now usually known by the name of its generalization - the FKG inequality). Talagrand started a line of works in search of quantitative versions of this fact by providing a lower bound on the correlation in terms of the influences of the functions. A famous conjecture of Chvátal was found by Friedgut, Kahn, Kalai and Keller to be equivalent to a related quantitative bound: If $f,g$ are increasing boolean functions and $g$ is antipodal (hence $g(x)=1-g(-x)$), then Cor(f,g) is at least $1/4$ times the minimal influence of $f$. We will discuss a new technique to prove such correlation inequalities. Motivated by this conjecture, we will show a strengthening of Talagrand's bound by giving estimates that also involve the second order Fourier coefficients of the functions.
  
  Abstract Video
  
  

  17:00 - 18:00

  Justin Salez, Université Paris-Dauphine

  Sparse expanders have negative curvature

  Abstract: We prove that bounded-degree expanders with non-negative Ollivier-Ricci curvature do not exist, thereby solving a long-standing open problem suggested by Naor and Milman and publicized by Ollivier (2010). In fact, this remains true even if we allow for a vanishing proportion of large degrees, large eigenvalues, and negatively-curved edges. To establish this, we work directly at the level of Benjamini-Schramm limits, and exploit the entropic characterization of the Liouville property on stationary random graphs to show that non-negative curvature and spectral expansion are incompatible "at infinity". We then transfer this result to finite graphs via local weak convergence. The same approach applies to the Bakry-Émery curvature condition CD(0,∞), thereby settling a recent conjecture of Cushing, Liu and Peyerimhoff (2019).
  
  Abstract Video Slides

   
              Contributed session

                  Session Chair: Wojciech Cygan, University of Wrocław, Wrocław

  
  

  18:20-18:40

  Artur Rutkowski, Wrocław University of Science and Technology

  Douglas identities for Sobolev-Bregman forms

  Abstract: Let $p>1$ and let $a^{\langle \kappa\rangle} = |a|^\kappa {\rm sgn}\, a$. Assume that $X_t$ is an isotropic Lévy process with the jumping kernel $\nu$. Under appropriate assumptions on $\nu$ and $D\subset \mathbb{R}^d$ we give the Douglas identity for Sobolev-Bregman forms: if $u$ is regular harmonic in $D$ with respect to $X_t$ and $g$ is the trace of $u$ on $D^c$, then \begin{align*}\iint_{(D^c\times D^c)^c} (u(x))^{\langle p-1\rangle} - u(y))^{\langle p-1\rangle})(u(x) - u(y)) \nu(x,y)\, dx \, dy\\ = \int_{D^c}\int_{D^c} (g(z)^{\langle p-1\rangle} - g(w)^{\langle p-1\rangle})(g(z) - g(w)) \gamma_D(z,w) \, dz \, dw.\end{align*} The kernel $\gamma_D$ is defined in terms of the Green function of $D$ for $X_t$: $$\gamma_D(z,w) = \int_D \int_D \nu(z,x)G_D(x,y)\nu(y,w) \, dx \, dy.$$ The proof relies on the Hardy-Stein identity of Bogdan, Dyda, and Luks. By extending it appropriately we also obtain a Douglas identity with a remainder term for non-harmonic functions $u$. The talk is based on a joint work with Krzysztof Bogdan, Tomasz Grzywny, and Katarzyna Pietruska-Pałuba.
  
  Abstract Video Slides
  
  

  18:40-19:00

  Bartłomiej Polaczyk, University of Warsaw

  From modified log-Sobolev inequalities to Beckner inequalities and moment estimates

  Abstract: We prove that in the context of general Markov semigroups Beckner inequalities with constants separated from zero as $p\to 1^+$ are equivalent to the modified log Sobolev inequality (previously only one implication was known to hold in this generality). Further, by adapting an argument by Boucheron et al. we derive Sobolev type moment estimates which hold under these functional inequalities. We illustrate our results with applications to concentration of measure estimates for various stochastic models, including random permutations, zero-range processes, strong Rayleigh measures, exponential random graphs, and geometric functionals on the Poisson path space. Joint work with R. Adamczak and M. Strzelecki.
  
  Abstract Video Slides
• 28May

     Probability models in mathematical physics

      Session Organizer: Tomasz Komorowski, IM PAS, Maria Curie-Skłodowska University

  
  
  15:00 - 16:00

  Bálint Tóth, Renyi Institute, University of Bristol

  Split-and-Merge in Stationary Random Stirring on Lattice Torus

  Abstract: We show that the cycle-length process of stationary random stirring (or, random interchange) on the $Z^d$-torus converges to the canonical Markovian split-and-merge process with the invariant (and reversible) measure given by the Poisson-Dirichlet law PD(1), as the size of the system grows to infinity. In the case of transient dimensions, $d \geq 3$, the problem is motivated by attempts to understand the onset of long-range order in the isotropic quantum Heisenberg ferromagnet, via random loop representations. Based on joint work with Dima Ioffe.
  
  Abstract Video Slides
  
  

  16:00 - 17:00

  Yu Gu, Carnegie Mellon University

  Gaussian fluctuations in surface growth models

  Abstract: The Kardar-Parisi-Zhang equation is a stochastic Hamilton-Jacobi equation that models the surface growth subjected to random perturbations. Understanding the statistical properties of the surface height function is of great interest from both mathematical and physical points of view. In recent years, there have been many fascinating developments in the one-dimensional setting. In this talk, I will present some work in high dimensions that proves a central limit theorem of the surface height function.
  
  Abstract Video Slides
  
  

  17:00 - 18:00

  Cedric Bernardin, Université Côte d’Azur

  Derivation of coupled KPZ-Burgers equation from multi-species zero-range processes

  Abstract: We consider the fluctuation fields of multi-species weakly-asymmetric zero-range interacting particle systems in one dimension, where the mass density of each species is conserved. Although such fields have been studied in systems with a single species, the multi-species setting is much less understood. Among other results, we show that, when the system starts from stationary states, with a particular property, the scaling limits of the multi-species fluctuation fields, seen in a characteristic traveling frame, solve a coupled Burgers SPDE, which is a formal spatial gradient of a coupled KPZ equation. Joint work with Tadahisa Funaki, Sunder Sethuraman.
  
  Abstract Video Slides