Searcharxiv⌕ Search

arXiv subjects

Jacek Zienkiewicz

Publications and source records attributed to Jacek Zienkiewicz.

At least 19 recordsLinked to original sources
arXiv2023

Dimension-free $L^p$ estimates for odd order maximal Riesz transforms in terms of the Riesz transforms

We prove a dimension-free $L^p(\mathbb{R}^d)$, $1<p<\infty$, estimate for the vector of maximal Riesz transforms of odd order in terms of the corresponding Riesz transforms. This implies a dimension-free $L^p(\mathbb{R}^d)$ estimate for the vector of maximal Riesz transforms in terms of the input function. We also give explicit estimates for the dependencies of the constants on $p$ when the order is fixed. Analogous dimension-free estimates are also obtained for single Riesz transforms of odd orders with an improved estimate of the constants. These results are a dimension-free extension of the work of J. Mateu, J. Orobitg, C. Pérez, and J. Verdera. Our proof consists of factorization and averaging procedures, followed by a non-obvious application of the method of rotations.

math.FA↗
arXiv2023

Dimension-free $L^p$ estimates for higher order maximal Riesz transforms in terms of the Riesz transforms

We prove a dimension-free $L^p(\mathbb{R}^d)$, $1<p<\infty$, estimate for the vector of higher order maximal Riesz transforms in terms of the corresponding Riesz transforms. This implies a dimension-free $L^p(\mathbb{R}^d)$ estimate for the vector of maximal Riesz transforms in terms of the input function. We also give explicit estimates for the dependencies of the constants on $p$ when the order is fixed. Analogous dimension-free estimates are also obtained for single higher order Riesz transforms with an improved estimate of the constants.

math.CA↗
arXiv2022

Whittle estimation based on the extremal spectral density of a heavy-tailed random field

We consider a strictly stationary random field on the two-dimensional integer lattice with regularly varying marginal and finite-dimensional distributions. Exploiting the regular variation, we define the spatial extremogram which takes into account only the largest values in the random field. This extremogram is a spatial autocovariance function. We define the corresponding extremal spectral density and its estimator, the extremal periodogram. Based on the extremal periodogram, we consider the Whittle estimator for suitable classes of parametric random fields including the Brown-Resnick random field and regularly varying max-moving averages.

math.ST↗
arXiv2020

Probabilistic approach to quantum separation effect for Feynman-Kac semigroup

Quantum tunnelling phenomenon allows a particle in Schrödinger mechanics tunnels through a barrier that it classically could not overcome. Even the infinite potentials do not always form impenetrable barriers. We discuss an answer to the following question: What is a critical magnitude of potential, which creates impenetrable barrier and for which the corresponding Schrödinger evolution system separates? In addition we describe some quantitative estimates for the separating effect in terms of cut-off potentials.

math.AP↗
arXiv2018

Affine stochastic equation with triangular matrices

We study solution X of the stochastic equation X = AX +B, where A is a random matrix and B,X are random vectors, the law of (A,B) is given and X is independent of (A,B). The equation is meant in law, the matrix A is 2x2 upper triangular, A_{11}=A_{22}>0, A_{12} is real. A sharp asymptotics of the tail of X =(X _1,X_2) is obtained. We show that under "so called" Kesten-Goldie conditions P (X_2>t)\sim t^{-a} and P (X_1>t )\sim t^{-a}(\log t)^b, where b =a or a\2.

math.PR↗
arXiv2017

Large global-in-time solutions to a nonlocal model of chemotaxis

We consider the parabolic-elliptic model for the chemotaxis with fractional (anomalous) diffusion. Global-in-time solutions are constructed under (nearly) optimal assumptions on the size of radial initial data. Moreover, criteria for blowup of radial solutions in terms of suitable Morrey spaces norms are derived.

math.AP↗
arXiv2017

Pointwise estimates for first passage times of perpetuity sequences

We consider first passage times $τ_u = \inf\{n:\; Y_n>u\}$ for the perpetuity sequence $$ Y_n = B_1 + A_1 B_2 + \cdots + (A_1\ldots A_{n-1})B_n, $$ where $(A_n,B_n)$ are i.i.d. random variables with values in ${\mathbb R} ^+\times {\mathbb R}$. Recently, a number of limit theorems related to $τ_u$ were proved including the law of large numbers, the central limit theorem and large deviations theorems. We obtain a precise asymptotics of the sequence ${\mathbb P}[τ_u = \log u/ρ]$, $ρ>0$, $u\to \infty $ which considerably improves the previous results. There, probabilities ${\mathbb P}[τ_u \in I_u]$ were identified, for some large intervals $I_u$ around $k_u$, with lengths growing at least as $\log\log u$. Remarkable analogies and differences to random walks are discussed.

math.PR↗
arXiv2016

Local criteria for blowup in two-dimensional chemotaxis models

We consider two-dimensional versions of the Keller--Segel model for the chemotaxis with either classical (Brownian) or fractional (anomalous) diffusion. Criteria for blowup of solutions in terms of suitable Morrey spaces norms are derived. Moreover, the impact of the consumption term on the global-in-time existence of solutions is analyzed for the classical Keller--Segel system.

math.AP↗
arXiv2015

Diffusion-driven blowup of nonnegative solutions to reaction-diffusion-ODE systems

In this paper we provide an example of a class of two reaction-diffusion-ODE equations with homogeneous Neumann boundary conditions, in which Turing-type instability not only destabilizes constant steady states but also induces blow-up of nonnegative spatially heterogeneous solutions. Solutions of this problem preserve nonnegativity and uniform boundedness of the total mass. Moreover, for the corresponding system with two non-zero diffusion coefficients, all nonnegative solutions are global in time. We prove that a removal of diffusion in one of the equations leads to a finite-time blow-up of some nonnegative spatially heterogeneous solutions.

math.AP↗
arXiv2015

Optimal criteria for blowup of radial and $N$-symmetric solutions of chemotaxis systems

A simple proof of concentration of mass equal to $8π$ for blowing up $N$-symmetric solutions of the Keller--Segel model of chemotaxis in two dimensions with large $N$ is given. Moreover, a criterion for blowup of solutions in terms of the radial initial concentrations, related to suitable Morrey spaces norms, is derived for radial solutions of chemotaxis in several dimensions. This condition is, in a sense, complementary to the one guaranteeing the global-in-time existence of solutions.

math.AP↗
arXiv2015

Existence of solutions for the Keller-Segel model of chemotaxis with measures as initial data

A simple proof of the existence of solutions for the two-dimensional Keller-Segel model with measures with all the atoms less than $8π$ as the initial data is given. This result has been obtained by Senba--Suzuki and Bedrossian--Masmoudi using different arguments. Moreover, we show a uniform bound for the existence time of solutions as well as an optimal hypercontractivity estimate.

math.AP↗
arXiv2014

Large deviation estimates for exceedance times of perpetuity sequences and their dual processes

In a variety of problems in pure and applied probability, it is of relevant to study the large exceedance probabilities of the perpetuity sequence $Y_n := B_1 + A_1 B_2 + \cdots + (A_1 \cdots A_{n-1}) B_n$, where $(A_i,B_i) \subset (0,\infty) \times {\mathbb R}$. Estimates for the stationary tail distribution of $\{ Y_n \}$ have been developed in the seminal papers of Kesten (1973) and Goldie (1991). Specifically, it is well-known that if $M := \sup_n Y_n$, then ${\mathbb P} \left\{ M > u \right\} \sim {\cal C}_M u^{-ξ}$ as $u \to \infty$. While much attention has been focused on extending this estimate, and related estimates, to more general processes, little work has been devoted to understanding the path behavior of these processes. In this paper, we derive sharp asymptotic estimates for the large exceedance times of $\{ Y_n \}$. Letting $T_u := (\log\, u)^{-1} \inf\{n: Y_n > u \}$ denote the normalized first passage time, we study ${\mathbb P} \left\{ T_u \in G \right\}$ as $u \to \infty$ for sets $G \subset [0,\infty)$. We show, first, that the scaled sequence $\{ T_u \}$ converges in probability to a certain constant $ρ> 0$. Moreover, if $G \cap [0,ρ] \not= \emptyset$, then ${\mathbb P} \left\{ T_u \in G \right\} u^{I(G)} \to C(G)$ as $u \to \infty$ for some "rate function" $I$ and constant $C(G)$. On the other hand, if $G \cap [0,ρ] = \emptyset$, then we show that the tail behavior is actually quite complex, and different asymptotic regimes are possible. We conclude by extending our results to the corresponding forward process, understood in the sense of Letac (1986), namely, the reflected process $M_n^\ast := \max\{ A_n M_{n-1}^\ast + B_n, 0 \}$ for $n \in {\mathbb N}$, where $M_0^\ast=0$.

math.PR↗
arXiv2013

A characterization of Hardy spaces associated with certain Schrödinger operators

Let $\{K_t\}_{t>0}$ be the semigroup of linear operators generated by a Schrödinger operator $-L=Δ- V(x)$ on $\mathbb R^d$, $d\geq 3$, where $V(x)\geq 0$ satisfies $Δ^{-1} V\in L^\infty$. We say that an $L^1$-function $f$ belongs to the Hardy space $H^1_L$ if the maximal function $\mathcal M_L f(x) = \sup_{t>0} |K_tf(x)|$ belongs to $L^1(\mathbb R^d) $. We prove that the operator $(-Δ)^{1\slash 2} L^{-1\slash 2}$ is an isomorphism of the space $H^1_L$ with the classical Hardy space $H^1(\mathbb R^d)$ whose inverse is $L^{1\slash 2} (-Δ)^{-1\slash 2}$. As a corollary we obtain that the space $H^1_L$ is characterized by the Riesz transforms $R_j=\frac{\partial}{\partial x_j}L^{-1\slash 2}$.

math.FA↗