In this talk, we report on joint work in progress with Yannick Sire (Johns Hopkins), where we study the Schrödinger equation

for the fractional Laplacian on Damek–Ricci spaces. More precisely, we obtain pointwise kernel estimates and deduce from them dispersive and Strichartz inequalities.

We plan to present a “multi-frequency Calderón-Zygmund analysis", introduced by Nazarov, Oberlin and Thiele and some applications. These results extend the classical theory. Indeed, it allows us to study a new kind of Calderón-Zygmund operators, mainly the sum of modulated Calderón-Zygmund operators.

Aiming to extend the classical results to these new operators, we first prove boundedness and then weighted boundedness. To do so, we make use of the abstract good-λ inequalities of Auscher and Martell, associated to a maximal sharp function (taking care of the “multi-frequency" framework).

This work is motivated by applications to the well-known Bochner-Riesz multipliers. Indeed, we will explain how such arguments allow us to get new weighted boundedness for such multipliers, involving Muckenhoupt’s weights. Moreover, this approach is very general and can be applied to generalised Bochner-Riesz multipliers (where the disc or the ball is replaced by another geometrical set).

This is a joint work with Zbigniew Palmowski (Wrocław University of Science and Technology) and Longmin Wang (Nankai University).

Let 0 < α < 2, d = 1,2,…, and let X = {X_t,t ≥ 0} be the isotropic α-stable Lévy process in ℝ^d. We denote by ℙ_x the law of the process starting from x ∈ ℝ^d. Let ∅≠Γ ⊂ ℝ^d be an arbitrary open Lipschitz cone with vertex at the origin 0. We define

(1)

the time of the first exit of X from Γ. The following measure μ will be called the Yaglom limit for X and Γ.

Theorem 1. There is a probability measure μ concentrated on Γ such that for every Borel set A ⊂ ℝ^d,

(2)

The result is proved in [2]. The above condition τ_Γ > t means that X stays, or survives, in Γ for time longer than t. Theorem 1 asserts that, given its survival, X_t rescaled by t^1∕α has a limiting distribution independent of the starting point. We note that rescaling is essential for the limit to be nontrivial. The Yaglom limit μ corresponds with the idea of “quasi-stationarity”, as expressed by Bartlett [1]:

It still may happen that the time to extinction is so long that it is still of more relevance to consider the effectively ultimate distribution (called a quasi-stationary distribution) [...]

We also construct and estimate entrance laws for the process from the vertex into the cone. Our approach relies on the scalings of the stable process and the cone, which allow to express the temporal asymptotics of the distribution of the process at infinity by means of the spatial asymptotics of harmonic functions of the process at the vertex; on the representation of the probability of survival of the process in the cone as a Green potential; and on the approximate factorization of the heat kernel of the cone, which secures compactness and yields a limiting (Yaglom) measure by means of Prokhorov’s theorem.

References

It is known that the product of two functions, one in H¹(ℝⁿ) and the other one in BMO(ℝⁿ), can be given a meaning and may be written as the sum of an integrable function and a function in H^log(ℝⁿ), which is defined as the space of functions f such that the non tangential maximal function f satisfies

It was natural to see what is the situation for the Hardy space H^p(ℝⁿ) when p < 1. For p ∈ (0,1) and = 1∕p - 1, let ^β(ℝⁿ) be the homogeneous Lipschitz space, which is the dual of H^p(ℝⁿ). While the pointwise product fg, with f ∈ H^p(ℝⁿ) and g ∈^β(ℝⁿ) does not make sense in general, it is possible to define it in the distribution sense and to prove that it belongs to the space L¹(ℝⁿ) + H_wp^p(ℝⁿ), with w_p the weight

This is in relation with multipliers of the (homogeneous) Lipschitz space ^α(ℝⁿ).

As an application, one has estimates for the div-curl product.

This is a joint work with Jun Cao, Luong Dang Ky, Liguang Liu, Dachun Yang and Wen Yuan.

Let Ω ⊆ ℝⁿ be open and A be a complex uniformly accretive matrix function on Ω. Consider the divergence-form operator L^A = -÷ (A∇) with Neumann boundary conditions in Ω. We show that the associated parabolic problem u^′(t) + L^Au(t) = f(t), u(0) = 0 has maximal regularity in L^p(Ω), for all p ∈ (1,+∞) such that A satisfies an algebraic condition called p-ellipticity. The given range of exponents is optimal for this class of operators.

The talk is based on a work in progress with Oliver Dragičević

One of the important features of the Heisenberg group is that it may be viewed as the boundary of a domain in ℂⁿ; in particular it has a CR (Cauchy–Riemann) structure. We examine other nilpotent groups with a CR structure and some of their properties

We introduce a condition on accretive matrix functions, called p-ellipticity, and discuss its applications to the L^p theory of elliptic PDE with complex coefficients. The examples we present concern:

1. generalized convexity of power functions (Bellman functions),
2. dimension-free bilinear embeddings,
3. L^p-contractivity of semigroups,
4. holomorphic functional calculus,
5. regularity theory of elliptic PDE with complex coefficients,
6. maximal L^p regularity for divergence-form operators with Neumann boundary conditions.

Example (5) is due to Dindoš and Pipher. Example (3) extends earlier theorems by Cialdea and Maz’ya.

The p-ellipticity condition arises from studying uniform positivity of a quadratic form associated with the matrix in question on one hand, and the Hessian of a power function on the other.

The talk is based on joint work with Andrea Carbonaro.

Let A and B be two n×n Hermitian matrices. Assume that the eigenvalues α₁,…,α_n of A are knowm, as well as the eigenvalues β₁,…,β_n of B. What can be said about the eigenvalues of the sum C = A + B ? This is Horn’s problem. In 1962 Horn proposed a conjecture, the so-called Horn’s conjecture, which says: the set of possible eigenvalues γ₁,…,γ_n for C is determined by a system of linear inequalities of the form

We revisit this problem from a probabilistic point of view. The set of Hermitian matrices X with spectrum {α₁,…,α_n} is an orbit _α for the natural action of the unitary group U(n): XUXU^* (U ∈ U(n)). Assume that the random Hermitian matrix X is uniformly distributed on the orbit _α, and the random Hermitian matrix Y is uniformly distributed on _β. In this talk we will present a formula for the joint distribution of the eigenvalues of the sum Z = X + Y . The proof involves orbital measures with their Fourier transforms, and Heckman’s measures.

We consider real-valued branching random walks and prove a large deviation result for the position of the rightmost particle. The position of the rightmost particle is the maximum of a dependent collection of a random number of random walks. We characterise the rate function as the solution of a variational problem. We consider the same random number of independent random walks, and show that the maximum of the branching random walk is dominated by the maximum of the independent random walks. For the maximum of independent random walks, we derive a large deviation principle as well. It turns out that the rate functions for upper large deviations coincide, but the rate functions for lower large deviations do not.

Based on joint work with Thomas Höfelsauer.

Let S_n = ∑ _i=1ⁿX_i be a sum of independent non-lattice random variables (X_i)_i≥1 under assumption Λ(s) = log E[e^sX₁] < +∞, for some s > 0, and let q = Λ′(s). Denote by Λ^* the Frenchel-Legendre transformation of Λ. Bahadur and Rao [Ann. Math. Stat., 31(1960), 1015-1027] and Petrov [Theory. Prob. Appl., 10(1965), 287-298] have established exact large deviation expansions of the followin form ℙ(S_n ≥ n(q + l_n)) ~ exp(-nΛ^*(s + l_n)) as l_n → 0 and n →∞. These milestone results have numerous applications in a variety of problems in pure and applied probability.

The goal is to prove equivalent expansions for the products of random matrices. Specifically, consider the product G_n := g_ng₁, where (g_n)_n≥1 is a sequence of i.i.d. d×d real random matrices of the same law μ. Assume that the support of μ is strongly irreducible and proximal for invertible matrices or allowable, contains at least one strictly positive matrix and is non-arithmetic for positive matrices. Let I_μ = {s ≥ 0 : E(∥g₁∥^s) < +∞}. Denote κ(s) = lim_n→∞ and Λ(s) = log κ(s), s ∈ I_μ. We prove large deviation expansions for the probability ℙ(log |G_nx|≥ n(q + l_n)) as n →∞, where x is a starting point on the unit sphere, q = Λ′(s) and l_n → 0. The asymptotics are expressed in terms of the eigenfunctions and invariant measures of the transfer operators P^s related to the Markov chain representation of log |G_nx| and log G_n^i,j.

A typical result is as follows. Denote by r_s the strictly positive eigenfunction of the transfer operator P^s corresponding to the eigenvalue κ(s) and by π_s the unique invariant probability measure of the normalized transfer operator P^s. Set also σ_s² = Λ′′(s), which, under the adopted assumptions, is a positive number.

Theorem 2. Let s ∈ I_μ^∘ and q = Λ′(s). Under appropriate moment assumptions, for any positive sequence (l_n)_n≥1 satisfying lim_n→∞l_n = 0, we have, uniformly in x on the unit sphere and |l|≤ l_n,

ℙ(log |Gnx|≥ n(q + l)) ~exp  as  n →∞.

Moreover, the rate function Λ^*(q + l) admits the following expansion: for l in a small neighborhood of 0,

Λ*(q + l) = Λ*(q) + sl + -ζs,

where ζ_s(t) is the Cramér series ζ_s(t) = ∑ _k=3^∞c_s,kt^k-3 = + O(t), m_3,s = Λ⁽³⁾(s).

A similar large deviation expansion is established for the entries of positive matrices, under the Kesten condition, where, for the proofs, we develop a spectral gap theory for a cocycle related the scalar product. These results are based on a joint work with Hui Xiao and Quansheng Liu.

Let T be a supercritical Galton-Watson tree endowed with a family of resistances (r(e)). Following Addario-Berry, Broutin and Lugosi (2009) and Lyons and Pemantle (1992), we are interested in the following two choices of (r(e)):

(i) r(e) = m^d(e)ξ(e) with m the mean number of offsprings of T, d(e) the height of the edge e and ξ(e) a family of i.i.d. positive random variables;

(ii) r(e) = ∏ _s≤eA(s) with a family (A(e)) of i.i.d. positive random variables.

We study the asymptotic behaviors of the effective conductances between the root and the vertices in the n-th generation of T, by way of a class of recursive equations on trees.

This talk is based on a joint work with Dayue Chen (Beijing University) and Shen Lin (Paris 6).

Let (P_r)_r∈ℕ be a collection of positive random variables satisfying ∑ _r≥1P_r = 1 a.s. Assume that, given (P_r)_r∈ℕ, ‘balls’ are allocated independently over an infinite collection of ‘boxes’ 1, 2,… with probability P_r of hitting box r, r ∈ ℕ. The occupancy scheme arising in this way is called the infinite occupancy scheme in the random environment (P_r)_r≥1.

A popular model of the infinite occupancy scheme in the random environment assumes that the probabilities (P_r)_r∈ℕ are formed by an enumeration of the a.s. positive points of

(3)

where X := (X(t))_t≥0 is a subordinator (a nondecreasing Lévy process) with X(0) = 0, zero drift, no killing and a nonzero Lévy measure, and ΔX(t) is a jump of X at time t. Since the closed range of the process X is a regenerative subset of [0,∞) of zero Lebesgue measure, one has ∑ _r≥1P_r = 1 a.s. When X is a compound Poisson process, collection (3) transforms into a residual allocation model

(4)

where W₁, W₂,… are i.i.d. random variables taking values in (0,1).

Next, I define a nested infinite sequence of the infinite occupancy schemes in random environments. This means that I construct a nested sequence of environments (random probabilities) and the corresponding ‘boxes’ so that the same collection of ‘balls’ is thrown into all ‘boxes’. To this end, I use a weighted branching process with positive weights which is nothing else but a multiplicative counterpart of a branching random walk.

The nested sequence of environments is formed by the weights (R(u))_|u|=1 = (P_r)_r∈ℕ, (R(u))_|u|=2,…, say, of the subsequent generations individuals in a weighted branching process. Further, I identify individuals with ‘boxes’. At time j = 0, all ‘balls’ are collected in the box ⊘ which corresponds to the initial ancestor. At time j = 1, given (R(u))_|u|=1, ‘balls’ are allocated independently with probability R(u) of hitting box u, |u| = 1. At time j = k, given (R(u))_|u|=1,…,(R(u))_|u|=k, a ball located in the box u with |u| = k is placed independently of the others into the box ur, r ∈ ℕ with probability R(ur)∕R(u).

Assume that there are n balls. For r = 1,2,…,n and j ∈ ℕ, denote by K_n,j,r the number of boxes in the jth generation which contain exactly r balls and set

where x⌈x⌉ = min{n ∈ ℤ : n ≥ x} is the ceiling function. With probability one the random function sK_n,j(s) is right-continuous on [0,1) and has finite limits from the left on (0,1] and as such belongs to the Skorokhod space D[0,1]. I am going to present sufficient conditions which ensure functional weak convergence of (K_n,j1(s),…,K_n,jm(s)), properly normalized and centered, for any finite collection of indices 1 ≤ j₁ < … < j_m as the number n of balls tends to ∞. If time permits, I shall discuss specializations of the general result to (P_r)_r∈ℕ given by (3) and (4).

The talk is based on a work in progress, joint with Sasha Gnedin (London).

We shall discuss several open (up to our knowledge) problems related to limit theorems for instantaneous functions of Markov chains with stable limits.

These include, among others:

• Lack of regularity assumptions in the Central Limit Theorem for ARCH(1)/GARCH(1,1) with infinite variation.
• Minimal form of compactness assumption imposed on the transition operator.
• What happens in the slot between geometric ergodicity and the L²-spectral gap property?

We extend finite-dimensional max-linear models to models on infinite graphs, and investigate their relations to classical percolation theory, more precisely to nearest neighbor bond percolation. We focus on the plane square lattice ℤ² with edges to the nearest neighbours, where we direct all edges in a natural way (to the right or up) resulting in a directed acyclic graph (DAG) on ℤ². On this infinite DAG a random sub-DAG may be constructed by choosing vertices and edges between them at random. In a Bernoulli bond percolation DAG edges are independently declared open with probability p ∈ (0,1) and closed otherwise. The random DAG consists then of the vertices and the open directed edges.

We find for the subcritical case where p ≤ 1∕2 that two random variables of the max-linear model become independent with probability 1, whenever their distance tends to infinity. In contrast, for the supercritical case where p > 1∕2 two random variables are dependent with positive probability, even when their node distance tends to infinity.

We also consider changes in the dependence properties of random variables on a sub-DAG H of a finite or infinite graph on ℤ², when enlarging this subgraph. The method of enlargement consists of adding nodes and edges of Bernoulli percolation clusters. Here we start with X_i and X_j independent in H, and answer the question, whether they can become dependent in the enlarged graph. We evaluate critical probabilities such that X_i and X_j become dependent in the enlarged graph with positive probability or with probability 1. We find in particular that for every DAG H with finite number of nodes, in the enlarged graph, X_i and X_j remain independent with positive probability. On the other hand, if H has nodes ℤ² and percolates everywhere; i.e. every connected component of H is infinite, then X_i and X_j become dependent with probability 1 in the enlarged graph.

[1] Gissibl, N. and Klüppelberg, C. (2018) Max-linear models on directed acyclic graphs. Bernoulli 24(4A), 2018, 2693–2720.

[2] Gissibl, N., Klüppelberg, C. and Otto, M. (2018) Tail dependence of recursive max-linear models with regularly varying noise variables. Econometrics and Statistics. To appear.

[3] Klüppelberg, C. and Sönmez, E. (2018) Max-linear models on infinite graphs generated by Bernoulli bond percolation. In preparation.

We consider a random variable R defined as a solution of the affine stochastic equation

RMR + Q R and (M,Q) independent.

Under suitable conditions, R may be represented as the series

h(x) = inf t≥1.

Moreover, we deal with the case of dependent M and Q and give asymptotic bounds for -log ℙ(R > x). It turns out that dependence structure between M and Q has a significant impact on the asymptotic rate of logarithmic tails of R. Such phenomenon is not observed in the case of heavy-tailed perpetuities. We show that

- c1 ≤ liminf x→∞ and limsupx→∞ ≤-c2,

where c₁ and c₂ are positive and explicit and

- d1 ≤ liminf x→∞ ≤ limsupx→∞ ≤-d2,

where d₁ and d₂ are explicit and optimal and

The spectral norm of any symmetric matrix is bigger than the largest Euclidean norm of its rows. We show that for Gaussian matrices with independent entries this obvious bound may be reversed in average up to a universal constant.

Theorem 3. Let X = (X_ij)_i,j≤d be a symmetric matrix such that (X_ij)_i≤j are independent centered Gaussian random variables. Then

We also discuss how to estimate quantities in the above theorem by an explicit expression in terms of the variances of the matrix entries and present similar result for Schatten norms.

Based on a joint work with Ramon van Handel (Princeton) and Pierre Youssef (Paris).

Grushin operators are a classic example of degenerate elliptic, sub-elliptic operators. The existence of a Mihlin–Hörmander functional calculus on L^p for such operators is known in great generality, but sharp results have been proved only in few instances. We report on recent progress.

References

Let L_n be the least common multiple of a random set of integers obtained from {1,2,…,n} by retaining each element with probability θ ∈ (0,1) independently of the others. In the talk I will discuss several asymptotic results for log L_n, as n →∞. In particular, it will be shown that if θ is fixed, then a functional limit theorem for the sequence of random functions [0,1] ∋ tlog L_⌊nt⌋ holds with a Gaussian limit process that is not a Brownian motion. I will also discuss regimes when θ varies with n leading to Poisson limit theorems. The talk is based on the joint paper [1] with G. Alsmeyer and Z. Kabluchko (Münster, Germany).

References

A celebrated result of C. Fefferman and E.M. Stein states that the Hardy space H¹(ℝⁿ) may be characterised in a number of ways, in particular, via Riesz transforms, the heat maximal operator and the Poisson maximal operator. It is natural to ask whther the same holds in a more general setting.

In this talk, we shall concentrate on Damek-Ricci spaces, introduced by Ewa Damek and Fulvio Ricci in the 90’s. We shall define Hardy spaces in terms of the Riesz transform, the heat maximal operator and the Poisson maximal operator. Quite surprisingly we show that these three spaces are mutually distinct. Relations with other Hardy-type spaces introduced by Mauceri, Meda and Vallarino will be discussed. Extensions to more general Riemannian manifolds will also be examined.

This is joint work with Alessio Martini and Maria Vallarino.

In various probabilistic models such as Pólya urns, search trees, and fragmentation processes, complex smoothing equations arise in limit theorems for quantities of interest.

In my talk, I will consider such complex smoothing equations and explain the relation to convergence of Biggins’ martingales in the branching random walk at complex parameters. For those martingales, Biggins (1992) proved local uniform convergence at complex parameters within a certain open domain. I will explain how critical smoothing equations are related to martingale convergence on the boundary of this domain. If time permits, I will also address rates of convergence.

The talk is based on joint work with Alexander Iksanov (Kyiv), Konrad Kolesko (Innsbruck) and Sebastian Mentemeier (Kassel).

This is joint work with Herold Dehling (Bochum), Muneya Matsui (Nagoya), Gennady Samorodnitsky (Cornell) and Laleh Tafakori (Melbourne). Distance covariance was introduced by Székely, Rizzo and Bakirov (2007) as a measure of dependence between vectors of possibly distinct dimensions. Since then it has attracted attention in various fields of statistics and applied probability. The distance covariance of two random vectors X,Y is a weighted L² distance between the joint characteristic function of (X,Y ) and the product of the characteristic functions of X and Y . It has the desirable property that it is zero if and only if X,Y are independent. This is in contrast to classical measures of dependence such as the correlation between two random variables: zero correlation corresponds to the absence of linear dependence but does not give any information about other kinds of dependencies. We consider the distance covariance for stochastic processes X,Y defined on some interval and having square integrable paths, including Lévy processes, fractional Brownian, diffusions, stable processes, and many more. Since distance covariance is defined for vectors we consider discrete approximations to X,Y . We show that sample versions of the discretized distance covariance converge to zero if and only if X,Y are independent. The sample distance covariance is a degenerate V -statistic and, therefore, has rate of convergence which is much faster than the classical -rates. This fact also shows nicely in simulation studies for independent X,Y in contrast to dependent X,Y .

Let X be a metric space with a doubling measure satisfying μ(B) ≳ r_Bⁿ for any ball B with any radius r_B > 0. Let L be a non negative selfadjoint operator on L²(X). We assume that e^-tL satisfies a Gaussian upper bound and that the flow e^itL satisfies a typical L¹ - L^∞ dispersive estimate of the form

Then we prove a similar L¹ - L^∞ dispersive estimate for a general class of flows e^itϕ(L), with ϕ(r) of power type near 0 and near ∞. In the case of fractional powers ϕ(L) = L^ν, ν ∈ (0,1), we deduce dispersive estimates for e^itLν with data in Sobolev, Besov or Hardy spaces H_L^p with p ∈ (0,1], associated to the operator L.

This is joint work with The Anh Bui, Piero D’Ancona and Xuan Thinh Duong.

In this lecture I plan to discuss some recent results obtained with E. Rela concerning Poincaré and Poincaré-Sobolev inequalities with weights. These results improve some classical estimates due to Fabes-Kenig-Serapioni obtained in the 80’s in connection with the local regularity of weak solutions of degenerate elliptic equations. I will also show that there is a connection with the Keith-Zhong phenomenon using extrapolation ideas.

Theorem 4. Given 1 ≤ p < n and w ∈ A_p we define p^* as the “degenerate" Poincaré-Sobolev exponent defined by

(5)

Then the following Poincarée-Sobolev inequality holds,

In this talk we explore the asymptotic enumeration of three-dimensional excursions confined to the positive octant. We focus on the critical exponent, which admits a universal formula in terms of the principal Dirichlet eigenvalue of a certain spherical triangle, itself being characterized by the steps of the model. Our main objective is to relate combinatorial properties of the step set (structure of the so-called group of the walk, existence of a Hadamard factorization, existence of differential equations satisfied by the generating functions) to geometric or analytic properties of the associated spherical triangle (remarkable angles, tiling properties, existence of an exceptional closed-form formula for the principal eigenvalue). As in general the eigenvalues of the Dirichlet problem on a spherical triangle are not known in closed form, we also develop a finite-elements method to compute approximate values, typically with a 10^-10 precision.

Let (G,K) be a Gelfand pair, with G of polynomial growth. The Gelfand spectrum Σ of L¹(K\G∕K) admits natural embeddings in some ℝ^k.

When G is a semidirect product G = K ⋉ H, bi-K-invariant functions on G are identified with K-invariant functions on H. It has been proved for many pairs of this kind, with H nilpotent, that the spherical transform is a bijection from the space of K-invariant Schwartz functions on H to restrictions to Σ of Schwartz functions on ℝ^k (Schwartz correspondence).

In this talk we look at Gelfand pairs in which K is contained in H and acts on it by inner automorphisms (K-central functions).

As a first step in the study of the Schwartz correspondence for these pairs,

1. we present some preliminary material, concerning spoherical analysis on sections of homogeneous bundles (joint work with A. Samanta),
2. we prove the Schwartz correspondence for the two-dimensional complex motion group H = U₂ ⋉ ℂ² (joint work with F. Astengo and B. Di Blasio).

Riesz distributions play an important role in the analysis on symmetric cones, tracing back to fundamental work by Gindikin in the 1970ies. In this talk, we study Riesz distributions and beta distributions in the Dunkl setting of type A, as well as an associated Laplace transform. These objects are closely related to a one-parameter generalization of the theory of hypergeometric functions of matrix argument which was introduced by Macdonald in the 1980ies, partly on a just formal level, and posted on the arviv [1] only some years ago.

We give the Dunkl-type Laplace transform a rigorous foundation and then focus on the question for which parameters Riesz and beta distributions in the Dunkl setting are actually measures. For Riesz distributions on symmetric cones, the admissible range of parameters is given by the so-called Wallach set. Analogous questions for beta distributions were recently studied in [2] in relation with Sonine formulas for Bessel functions on symmetric cones. In the Dunkl case, a natural generalization of the Wallach set enters the picture, and the question when beta distributions are actually measures is closely related to the existence of Sonine-type integral representations between Dunkl kernels with different multiplicities. As a consequence, we obtain examples where the intertwining operator between Dunkl operators associated with multiplicities k ≥ 0 and k^′≥ k is not positive, which disproves a long-standing conjecture.

Part of this talk is based on joint work with Michael Voit.

References:

[1] I.G. Macdonald, Hypergeometric Functions I. ArXiv 1309.4568.

[2] M. Rösler, M. Voit, Beta distributions and Sonine integrals for Bessel functions on symmetric cones. ArXiv 1801.07304. To appear in Stud. Appl. Math.

An Ornstein-Uhlenbeck operator in ℝⁿ is linear and elliptic with constant second-order coefficients. But the first-order coefficients are linear in the coordinates and such that they cause a drift inwards. This operator generates a semigroup, and we study the corresponding maximal operator. The relevant measure here is a gaussian measure, which replaces Lebesgue measure. Assuming only that the semigroup is normal, i.e., commutes with its adjoint, we prove that the maximal operator is of weak type (1,1) for the gaussian measure. This extends earlier results by several authors. The first step in the proof is a transformation of variables which gives the semigroup a reasonable, explicit expression.

The talk is concerned with the problem of ergodicity for iterated function systems consisting of homeomorphisms on the interval or circle. It will be also shown that such systems, under quite general conditions, satisfy the Central Limit Theorem and Law of the Iterated Logarithm.

The topic of dimension-free L^p estimates for Hardy–Littlewood maximal functions over convex bodies in ℝ^d had been mainly developed in the 80’ and 90’ in the work of Stein, Bourgain, Carbery, and Müller. The interest in the topic has been recently renewed due to recent progress by Aldaz 2011 and Bourgain 2014. However, up to now, nothing has been done in the discrete context, i.e. when ℝ^d is replaced by ℤ^d.

In this talk we present first dimension-free results for discrete Hardy–Littlewood maximal functions. First we give an example showing that the phenomenon is not as robust as in the continuous case. Then we focus on the case of the discrete cube. New estimates for the Fourier transform of the characteristic function of the discrete cube are crucial to our work. An important ingredient of our approach are also the methods developed in our previous work, where we proved dimension-free estimates for r-variations of Hardy–Littlewood averaging operators on ℝ^d.

The talk is based on joint work with J. Bourgain, M. Mirek, and E.M. Stein.

Sendov et Shen [1] studied the class of Bernstein functions whose Lévy measures admit a harmonic convex density. We give several equivalent definitions and show that this class produces refined properties for the finite-dimensional distributions of the associated subordinators. We also propose a generalization of the class of Sendov and Chen and an answer to the open problem, strongly connected with stable subordinators, raised by them in [1] .

[1] Sendov, H., Shan, S.: New Representation Theorems for Completely Monotone and Bernstein Functions with Convexity Properties on Their Measures. J Theor Probab, Volume 28, Issue 4, pp 1689–1725, 2015.

Let γ_-1 be the absolutely continuous measure on ℝⁿ whose density is the reciprocal of a Gaussian and consider the weighted Laplacian which is self-adjoint on L²(γ_-1). This operator may be seen as a restriction of the Laplace–Beltrami operator on a warped-product manifold whose Ricci tensor is unbounded from below. In this talk, I will present boundedness and unboundedness results for the purely imaginary powers and the first order Riesz transforms associated with the operators + λI, λ ≥ 0, from some new Hardy-type spaces adapted to γ_-1 to L¹(γ_-1).

In 1984, A. Hulanicki showed that the spectral calculus associated with a positive Rockland operator on a graded group maps Schwartz multipliers into Schwartz convolution kernels. In 2010, A. Martini generalized the preceding result to more general families of differential operators including the commutative finite families of homogeneous left-invariant differential operators one of which is Rockland.

Focusing on the setting of stratified Lie groups, I will consider:

• a converse of the aforementioned result, that is: given a Schwartz kernel, does it correspond to a Schwartz multiplier?
• an analogue of the Riemann–Lebesgue lemma: given an integrable kernel, does it correspond to a continuous multiplier?

I will recall and discuss a notion of heat content related to Lévy processes in Euclidean spaces. To start with, I will present instructive examples including Brownian motion and stable processes, and next I will focus on the study of the small time behaviour of the heat content (and of its more general version) for a rich class of Lévy processes.
The talk is based on the joint project with Tomasz Grzywny (Wrocław University of Science and Technology).

We will consider a random walk on integers in site-dependent random environment. We will show how to use the associated branching process to obtain a precise large deviation asymptotic for the random walk. The talk is based on a joint work with Dariusz Buraczewski.

In this talk we will discuss localisation of Bochner Riesz means of order 0 for the sub-Laplacian on the Heisenberg Group ℍ^d. More precisely, generalising the method of Carbery and Soria (1988) in the context of ℍ^d, we shall show that for any β > 0 and 0 < η < 1, lim_R→∞R^-βS_Rf(z,t) goes to 0 a.e. on the set {(z,t) ∈ ℍ^d : ∥(z,t)∥≤ 1} for f ∈ L²(ℍ^d \{∥(z,t)∥≤ 3},∥(z,t)∥^-ηdzdt), where S_R are Bochner Riesz means of order 0 for on ℍ^d.

This is joint work with Jotsaroop Kaur.

Let be the Laplace operator on ℝ^d, d ≥ 3 or the Laplace Beltrami operator on the harmonic NA group. We give necessary and sufficient conditions for the existence of entire bounded or large solutions of the equation

On ℝⁿ let f_Δ^#(x) and M^Δf(x) denote the classical dyadic sharp function and dyadic maximal function respectively, that is,

fΔ#(x) = sup x∈Q∈Δ∫ Q|f(y) - fQ|dy, MΔf(x) = sup x∈Q∈Δ∫ Q|f(y)|dy,

where Δ denotes the collection of all dyadic cubes in ℝⁿ and f_Q is the average of f over Q. Suppose that f ∈ L^p₀(ℝⁿ) for some p₀. The well-known Fefferman-Stein inequality asserts that if 1 < p < ∞, 1 ≤ p₀ ≤ p, and f_Δ^# ∈ L^p(ℝⁿ), then M^Δf ∈ L^p(ℝⁿ) and

(6)

Let Ω be a domain in ℝⁿ. Our goal is to define for f ∈ L_loc¹(Ω) a localized version f_loc^# of the sharp function which will satisfy the analogue of the Fefferman–Stein inequality. By localized we mean that the cubes which are taken in the definition of f_loc^#(x) are contained in a bounded set _x ⊂ Ω. So one possible definition can be taken as follows. Let τ : Ω → (0,∞). For f ∈ L_loc¹(Ω) we set

This talk is based on joint work with Jacek Dziubański.

The boundary Harnack inequality is a statement about positive harmonic functions in an open set D, which are equal to zero on a part of the boundary. It states that if D is regular enough, z is a boundary point of D, f and g are positive and harmonic in D, and both f and g converge to 0 on ∂D ∩ B(z,R), then for every r ∈ (0,R)

supx∈D∩B(z,r)

≤ cBHI inf x∈D∩B(z,r),

(7)

where constant c_BHI depends only on D and r.

Over last 20 years BHI was shown to hold for various types of stochastic processes and domains. What often followed was the existence of the limits of the ratios of harmonic functions as r → 0 in (7) and their explicit decay rate.

Results that are presented in my talk are about the decay rate of f next to the boundary of D for non-symmetric strictly α-stable Lévy processes. We assume that the Lévy measure of X has a density function which is Hölder continuous on the unit sphere with exponent ϵ and for domains D of C^1,1 class for α < 1 and of C^2,α+ϵ-1 class for 1 ≤ α < 2.

Theorem 5. Let z ∈ ∂D ∩ B(z,R₀) and let f be a non-negative function which is harmonic in D ∩ B(z,R₀) and vanishes continuously on D^c ∩ B(z,R₀), then

lim exists as x → z,x ∈ D,

where β(x) = αℙ⁰(⟨X_t,n(x)⟩ > 0), vector n(x) is a normal vector to the boundary of D at the point z(x) — orthogonal projection of x onto the boundary of D.

The classical Hörmader multiplier theorem states that if m satisfies

(S)

with some β > d∕2, then operator m(-Δ) is of weak type (1,1) and bounded on L^p(R^d) for every p ∈ (1,∞). Here 0 ≤ η ∈ C_c^∞(2^-1,2) is fixed and W^2,β(R) is the standard L²-Sobolev norm on R. It is well-known that the constant d∕2 is sharp. Similar multiplier theorem is known on the Hardy space H¹(ℝ^d). In the talk we shall consider some generalizations of this results.

Suppose the metric-measure space (X,ρ,μ) satisfies the doubling condition, which implies that there exists constant d such that

(D)

Assume that the semigroup P_t = exp(-tA) generated by a self-adjoint positive operator A satisfies the lower and upper gaussian bounds. Moreover, suppose that there exists C > 0, such that for every R > 0 and a measurable function m on R such that supp(m) ⊆ [R∕2,R] we have

(P)

We prove that if the bounded function m satisfies (S) with β > d∕2, then m(A) is bounded from H¹(A) to H¹(A), where H¹(A) = {f ∈ L¹(X) : ∥sup_t>0|P_tf|∥_L¹(X,μ) < ∞} denotes the Hardy space associated with A.

As a main example we consider the multiparameter Bessel operator B = B₁ + ... + B_N, where B_if(x) = -δ_j²f(x) - α_j∕x_jδ_jf(x), α_j > -1, on the space (0,∞)^N with the Euclidean metric and the measure x^α₁...x_N^α_Ndx₁...dx_N. The condition (D) is satisfied with d = ∑_j=1^N max(1,1 + α_j). By analysing the imaginary powers B^ib,b ∈ ℝ, of the one-dimensional Bessel operator we show that in this case the constant d∕2 is sharp.

The talk is based on the joint work with Marcin Preisner (University of Wrocław).

In joint work with Mateusz Kwaśnicki we discuss representation of certain functions of the Laplace operator Δ as Dirichlet-to-Neumann maps for appropriate elliptic operators in half-space. A classical result identifies (-Δ)^1∕2, the square root of the d-dimensional Laplace operator, with the Dirichlet-to-Neumann map for the (d + 1)-dimensional Laplace operator Δ_t,x in (0,∞) × ℝ^d. Caffarelli and Silvestre extended this to fractional powers (-Δ)^α∕2, which correspond to operators ∇_t,x(t^1-α∇_t,x). We provide an analogous result for all complete Bernstein functions of (-Δ) using Krein’s spectral theory of strings.

We give sharp two-sided estimates of the transition probability densities for the Brownian motion on a M-complex of size M ∈ ℤ for a class of planar simple nested fractals.

Theorem 6. The transition probability densities g_M(t,x,y) for the Brownian motion reflecting in the vertices of an M-complex satisfy the following inequalities

where

fc(t,r) =	t-ds∕2 exp
hc(t,M) =	L-dfMdf-dw∕ exp

and constants c₁,...,c₆ do not depend on t, x, y or M.

A Hardy’s type inequality for Laguerre functions of Hermite type with the index α ∈ ({-1∕2}∪ [1∕2,∞))^d, is proved in the multi-dimensional setting with the exponent 3d∕4. For this purpose we uniformly estimate the L² norms associated with derivatives of an appropriate family of L² contractions. Moreover, we obtain the sharp analogue of Hardy’s inequality with L¹ norm replacing H¹ norm at the expense of increasing the exponent by an arbitrarily small value.

We present an extension theorem related to spaces generated by quadratic forms

Here D is an open subset of ℝ^d, and ν is a Lévy-type kernel. To establish our results, we use objects and methods from the potential theory, in particular the communication kernel

where P_D is the Poisson kernel of D.

Our extension theorem states that if squared increments of a function g given on D^c are integrable with weight γ_D, then the harmonic function, given for x ∈ D by the Poisson integral ∫ _D^cg(z)P_D(x,z)dz, serves as the extension of g, i.e. its _D form is finite.

We also provide estimates for the kernel γ_D and applications to the Dirichlet problem for nonlocal operators associated with ν.

The talk is based on joint work [1] with Krzysztof Bogdan, Tomasz Grzywny and Katarzyna Pietruska-Pałuba.

References

In this talk, we introduce a notion of discrete subordination and we consider a large class of subordinate random walks on the d-dimensional integer lattice via subordinators with Laplace exponents which are complete Bernstein functions satisfying certain scaling conditions at zero. We establish estimates for one-step transition probabilities, the Green function and the Green function of a ball, and prove the Harnack inequality for non-negative harmonic functions.

We consider the real hyperbolic space ℍⁿ = {x ∈ ℝⁿ : x_n > 0}, where the Laplace-Beltrami operator is given by

Most of the result have been obtained due to the probabilistic approach exploiting hyperbolic Brownian motion, which is a process generated by Δ_ℍⁿ. Its relationships to Bessel processes and Brownian motion killed by a suitable potential have played a crucial role at many points.

Entangled multilinear singular integral forms have been studied by several authors over the last ten years. They recently found applications in ergodic theory [5], in arithmetic combinatorics [4], to stochastic integration [9], and within the harmonic analysis itself [6]. Therefore, it would be useful to have a reasonably general theory establishing (or characterizing) L^p bounds for these objects. As a step in this program we take a result of Kovač [7], where the forms are dyadic and indexed by bipartite graphs, and generalize it to r-partite r-uniform hypergraphs. Furthermore, we also combine it with the techniques of Durcik [1],[2] in order to cover general translation-invariant Calderón-Zygmund kernels. Some higher-dimensional instances were already discussed by Kovač [8] and Durcik [3], but our hypergraph generalization prefers a combinatorial description of the structure over a geometric one. Consequently, we can study less symmetric entangled forms and show their estimates in an open range of L^p spaces.

This is a joint work with Vjekoslav Kovač (University of Zagreb).

References

We study the solution W to the stochastic equation WAW + B, where the distribution of (A,B) ∈ M_d×d(ℝ₊) × ℝ₊^d is given and it satisfies some integrability assumptions.

It was proved by Kesten that if the matrix Aⁿ has all entries positive for some n, then each coordinate W_i of the solution has a regularly varying tail, with a common tail index. On the other hand, if A is a diagonal matrix, the tail indices clearly can be different for each coordinate. It is natural to ask, what happens in non-trivial cases that do not fit into Kesten’s setting.

We assume that A is a triangular matrix. It does not satisfy the Kesten’s assumptions since Aⁿ is also triangular. We develop methods to prove that

The tail indices _i depend on the laws of the diagonal entries A_jj with j ≥ i and on positions of zero entries of the matrix A. They are given by an exact expression. The constants c_i are also calculated.

The talk is based on a joint work with Muneya Matsui.

Let μ be a probability measure on the real line having all of the moments finite. The orthogonal polynomial ensemble of size n ∈ ℕ is a measure on ℝⁿ proportional to

The associated linear statistics are expressions of the form

where f : ℝ → ℝ is a smooth function, x₀ is a real number and α ∈ (0,1).

We are going to present CLT for linear statistics under conditions imposed on the three-term reccurrence relation satisfied by the orthogonal polynomials associated with the measure μ. Following recent work of Gaultier Lambert, the proof is reduced to the derivation of precise asymptotics of the orthogonal polynomials in question.

It is a joint work with Bartosz Trojan.

We construct the fundamental solution (the heat kernel) p^κ to the equation ∂_t = ^κ, where under certain assumptions the operator ^κ takes one of the following forms,

κf(x)	:= ∫ ℝd(f(x + z) - f(x) - 1|z|<1)κ(x,z)J(z)dz,
κf(x)	:= ∫ ℝd(f(x + z) - f(x))κ(x,z)J(z)dz,
κf(x)	:= ∫ ℝd(f(x + z) + f(x - z) - 2f(x))κ(x,z)J(z)dz.

In particular, J : ℝ^d → [0,∞] is a Lévy density, i.e., ∫ _ℝ^d(1 ∧|x|²)J(x)dx < ∞. The function κ(x,z) is assumed to be Borel measurable on ℝ^d × ℝ^d satisfying 0 < κ₀ ≤ κ(x,z) ≤ κ₁, and |κ(x,z) - κ(y,z)|≤ κ₂|x - y|^β for some β ∈ (0,1). We prove the uniqueness, estimates, regularity and other qualitative properties of p^κ. It gives rise to a Feller semigroup (P_t)_t≥0 and a Feller process X = (X_t, ℙ^x) that in turn solves the martingale problem for (^κ,C_c^∞(ℝ^d)).

The talk is based on a joint work with T. Grzywny.

Differential subordination of real-valued martingales together with the basic properties was discovered by Burkholder in 1984. In this talk we will discuss weak differential subordination of martingales, which is a generalization of differential subordination to the infinite dimensional setting, and provide extension of the corresponding L^p estimates for vector-valued martingales. Also we will show more general estimates for weakly differentially subordinated martingales under the orthogonality assumption.

As a corollary we extend the results of Bañuelos and Bogdan (2007) and Bañuelos, Bielaszewski, and Bogdan (2011) on sharp estimates for the norms of a broad class of even Fourier multipliers (including e.g. second order Riesz transforms) to infinite dimensions, and provide new estimates for the norm of the Hilbert transform acting on general Banach space-valued functions.

The talk is partly based on joint work with Adam Osekowski (University of Warsaw).