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Kamil Kaleta

Publications and source records attributed to Kamil Kaleta.

At least 19 recordsLinked to original sources
arXiv2026

Critical discrete Hardy inequalities in $L^p$

We establish ground-state representations and critical Hardy inequalities in the discrete $L^p$ setting. In particular, we construct critical Hardy weights for the discrete Dirichlet Laplacian on the half-line and the discrete fractional Laplacian on the integers for all $p\in(1,\infty)$. Our approach uses Sobolev--Bregman forms, for which the ground-state representations are exact identities and the corresponding Hardy weights are expressed through linear operators acting on powers of positive superharmonic functions.

math.CA↗
arXiv2026

IDS for subordinate Brownian motions in Poisson random environment on nested fractals

We establish the Lifshitz singularity of the integrated density of states (IDS) for random Schr\"odinger operators \[ H^{\omega} = \phi(-\mathcal{L}) + V^{\omega} \] on planar unbounded nested fractals with the Good Labeling Property. Here, $\mathcal{L}$ is the Laplacian on the fractal, $\phi$ is an operator monotone function with mild regularity, and $V^{\omega}$ is a Poissonian random potential with a sufficiently regular profile. The main novelty of our work lies in showing that the study of $V^{\omega}$ can be effectively reduced to the analysis of certain alloy-type potential, where the sites are no longer lattice points as in the classical $\mathbb{Z}^d$ case, but fractal complexes. This observation enables us to apply an approach, new in the setting of Poissonian random fields, which allows us to treat a broad class of Bernstein functions $\phi$. In particular, it covers the case $\phi(\lambda)=(\lambda+m^{d_w/\vartheta})^{\vartheta/d_w}-m$, $\vartheta \in (0,d_w)$, $m>0$, corresponding to relativistic models, which were previously unattainable on fractals by known methods.

math.PR↗
arXiv2026

Smoothing on $L^1$ for ground state transformed semigroups in non-local settings

We study the $L^1$-smoothing properties for a broad class of semigroups arising from the ground state transformation of Schr\"odinger semigroups with confining potentials associated with non-local L\'evy operators, for which (asymptotic) ultracontractivity and hypercontractivity fail. Our work is inspired by Talagrand's convolution conjecture in the discrete cube setting, as well as by subsequent developments on the classical Ornstein--Uhlenbeck semigroup. The estimates we provide exhibit a clear dependence on the potential and the L\'evy measure defining the kinetic term operator, and they yield a description of the semigroups' action on $L^1$ in terms of Orlicz spaces. Our framework is quite general, encompassing fractional and relativistic Laplacians as kinetic operators. The results are illustrated by numerous examples demonstrating that the $L^1$-regularizing effects become stronger as $t \uparrow \infty$.

math.FA↗
arXiv2025

Heat kernels, intrinsic contractivity and ergodicity of discrete-time Markov chains killed by potentials

We study discrete-time Markov chains on countably infinite state spaces, which are perturbed by rather general confining (i.e.\ growing at infinity) potentials. Using a discrete-time analogue of the classical Feynman--Kac formula, we obtain two-sided estimates for the $n$-step heat kernels $u_n(x,y)$ of the perturbed chain. These estimates are of the form $u_n(x,y)\asymp \lambda_0^n\phi_0(x)\widehat\phi_0(y)+F_n(x,y)$, where $\phi_0$ (and $\widehat\phi_0$) are the (dual) eigenfunctions for the lowest eigenvalue $\lambda_0$; the perturbation $F_n(x,y)$ is explicitly given, and it vanishes if either $x$ or $y$ is in a bounded set. The key assumptions are that the chain is uniformly lazy and that the \enquote{direct step property} (DSP) is satisfied. This means that the chain is more likely to move from state $x$ to state $y$ in a single step rather than in two or more steps. Starting from the form of the heat kernel estimate, we define the intrinsic (or ground-state transformed) chains and we introduce time-dependent ultracontractivity notions -- asymptotic and progressive intrinsic ultracontractivity -- which we can link to the growth behaviour of the confining potential; this allows us to consider arbitrarily slow growing potentials. These new notions of ultracontractivity also lead to a characterization of uniform (quasi-)ergodicity of the perturbed and the ground-state transformed Markov chains. At the end of the paper, we give various examples that illustrate how our findings relate to existing models, e.g.\ nearest-neighbour walks on infinite graphs, subordinate processes or non-reversible Markov chains.

math.PR↗
arXiv2025

Decay of resolvent kernels and Schr\"odinger eigenstates for L\'evy operators

We study the spatial decay behaviour of resolvent kernels for a large class of non-local L\'evy operators and bound states of the corresponding Schr\"odinger operators. Our findings naturally lead us to proving results for L\'evy measures, which have subexponential or exponential decay, respectively. This leads to sharp transitions in the the decay rates of the resolvent kernels. We obtain estimates that allow us to describe and understand the intricate decay behaviour of the resolvent kernels and the bound states in either regime, extending findings by Carmona, Masters and Simon for fractional Laplacians (the subexponential regime) and classical relativistic operators (the exponential regime). Our proofs are mainly based on methods from the theory of operator semigroups.

math.SP↗
arXiv2025

Heat kernels and Green functions for fractional Schr\"{o}dinger operators with confining potentials

We give two-sided, global (in all variables) estimates of the heat kernel and the Green function of the fractional Schr\"odinger operator with a non-negative and locally bounded potential $V$ such that $V(x) \to \infty$ as $|x| \to \infty$. We assume that $V$ is comparable to a radial profile with the doubling property. Our bounds are sharp with respect to spatial variables and qualitatively sharp with respect to time. The methods we use combine probabilistic and analytic arguments. They are based on the strong Markov property and the Feynman--Kac formula.

math.PR↗
arXiv2023

Quasi-ergodicity of compact strong Feller semigroups on $L^2$

We study the quasi-ergodicity of compact strong Feller semigroups $U_t$, $t > 0$, on $L^2(M,\mu)$; we assume that $M$ is a locally compact Polish space equipped with a locally finite Borel measue $\mu$. The operators $U_t$ are ultracontractive and positivity preserving, but not necessarily self-adjoint or normal. We are mainly interested in those cases where the measure $\mu$ is infinite and the semigroup is not intrinsically ultracontractive. We relate quasi-ergodicity on $L^p(M,\mu)$ and uniqueness of the quasi-stationary measure with the finiteness of the heat content of the semigroup (for large values of $t$) and with the progressive uniform ground state domination property. The latter property is equivalent to a variant of quasi-ergodicity which progressively propagates in space as $t \uparrow \infty$; the propagation rate is determined by the decay of $U_t \mathbb{1}_M(x)$. We discuss several applications and illustrate our results with examples. This includes a complete description of quasi-ergodicity for a large class of semigroups corresponding to non-local Schr\"odinger operators with confining potentials.

math.FA↗
arXiv2023

Integral kernels of Schrödinger semigroups with nonnegative locally bounded potentials

We give the upper and the lower estimates of heat kernels for Schrödinger operators $H=-Δ+V$, with nonnegative and locally bounded potentials $V$ in $\mathbb{R}^d$, $d \geq 1$. We observe a factorization: the contribution of the potential is described separately for each spatial variable, and the interplay between the spatial variables is seen only through the Gaussian kernel - optimal in the lower bound and nearly optimal in the upper bound. In some regimes we observe the exponential decay in time with the rate corresponding to the bottom of the spectrum of $H$. Our estimates identify in a fairly informative and uniform way the dependence of the potential $V$ and the dimension $d$. The upper estimate is more local; it applies to general potentials, including confining and decaying ones, even if they are non-radial, and mixtures. The lower bound is specialized to confining case, and the contribution of the potential is described in terms of its radial upper profile. Our results take the sharpest form for confining potentials that are comparable to radial monotone profiles with sufficiently regular growth - in this case they lead to two-sided qualitatively sharp estimates. In particular, we describe the large-time behaviour of nonintrinsically ultracontractive Schrödinger semigroups - this problem was open for a long time. The methods we use combine probabilistic techniques with analytic ideas. We propose a new effective approach which leads us to short and direct proofs.

math.FA↗
arXiv2023

Density of states for the Anderson model on nested fractals

We prove the existence and establish the Lifschitz singularity of the integrated density of states for certain random Hamiltonians $H^ω=H_0+V^ω$ on fractal spaces of infinite diameter. The kinetic term $H_0$ is given by $ϕ(-\mathcal L),$ where $\mathcal L$ is the Laplacian on the fractal and $ϕ$ is a completely monotone function satisfying some mild regularity conditions. The random potential $V^ω$ is of alloy-type.

math.SP↗
arXiv2022

Bound states and heat kernels for fractional-type Schrödinger operators with singular potentials

We consider non-local Schrödinger operators $H=-L-V$ in $L^2(\mathbf{R}^d)$, $d \geq 1$, where the kinetic terms $L$ are pseudo-differential operators which are perturbations of the fractional Laplacian by bounded non-local operators and $V$ is the fractional Hardy potential. We prove pointwise estimates of eigenfunctions corresponding to negative eigenvalues and upper finite-time horizon estimates for heat kernels. We also analyze the relation between the matching lower estimates of the heat kernel and the ground state near the origin. Our results cover the relativistic Schrödinger operator with Coulomb potential.

math.FA↗
arXiv2022

Exponential densities and compound Poisson measures

We prove estimates at infinity of convolutions $f^{n\star}$ and densities of the corresponding compound Poisson measures for a class of radial decreasing densities on $\mathbb{R}^d$, $d \geq 1$, which are not convolution equivalent. Existing methods and tools are limited to the situation in which the convolution $f^{2\star}(x)$ is comparable to initial density $f(x)$ at infinity. We propose a new approach which allows one to break this barrier. We focus on densities which are products of exponential functions and smaller order terms -- they are common in applications. In the case when the smaller order term is polynomial estimates are given in terms of the generalized Bessel function. Our results can be seen as the first attempt to understand the intricate asymptotic properties of the compound Poisson and more general infinitely divisible measures constructed for such densities.

math.PR↗
arXiv2022

On directional convolution equivalent densities

We propose a definition of directional multivariate subexponential and convolution equivalent densities and find a useful characterization of these notions for a class of integrable and almost radial decreasing functions. We apply this result to show that the density of the absolutely continuous part of the compound Poisson measure built on a given density $f$ is directionally convolution equivalent and inherits its asymptotic behaviour from $f$ if and only if $ f$ is directionally convolution equivalent. We also extend this characterization to the densities of more general infinitely divisible distributions on $\mathbb{R}^d$, $d \geq 1$, which are not pure compound Poisson.

math.PR↗
arXiv2022

Heat kernels of non-local Schrödinger operators with Kato potentials

We study heat kernels of Schrödinger operators whose kinetic terms are non-local operators built for sufficiently regular symmetric Lévy measures with radial decreasing profiles and potentials belong to Kato class. Our setting is fairly general and novel -- it allows us to treat both heavy- and light-tailed Lévy measures in a joint framework. We establish a certain relative-Kato bound for the corresponding semigroups and potentials. This enables us to apply a general perturbation technique to construct the heat kernels and give sharp estimates of them. Assuming that the Lévy measure and the potential satisfy a little stronger conditions, we additionally obtain the regularity of the heat kernels. Finally, we discuss the applications to the smoothing properties of the corresponding semigroups. Our results cover many important examples of non-local operators, including fractional and quasi-relativistic Schrödinger operators.

math.AP↗
arXiv2021

Decay of harmonic functions for discrete time Feynman--Kac operators with confining potentials

We propose and study a certain discrete time counterpart of the classical Feynman--Kac semigroup with a confining potential in countable infinite spaces. For a class of long range Markov chains which satisfy the direct step property we prove sharp estimates for functions which are (sub-, super-)harmonic in infinite sets with respect to the discrete Feynman--Kac operators. These results are compared with respective estimates for the case of a nearest-neighbour random walk which evolves on a graph of finite geometry. We also discuss applications to the decay rates of solutions to equations involving graph Laplacians and to eigenfunctions of the discrete Feynman--Kac operators. We include such examples as non-local discrete Schrödinger operators based on fractional powers of the nearest-neighbour Laplacians and related quasi-relativistic operators. Finally, we analyse various classes of Markov chains which enjoy the direct step property and illustrate the obtained results by examples.

math.PR↗
arXiv2019

Lifshitz tail for continuous Anderson models driven by Lévy operators

We investigate the behavior near zero of the integrated density of states $\ell$ for random Schrödinger operators $Φ(-Δ) + V^ω$ in $L^2(\mathbb R^d)$, $d \geq 1$, where $Φ$ is a complete Bernstein function such that for some $α\in (0,2]$, one has $ Φ(λ) \asymp λ^{α/2}$, $λ\searrow 0$, and $V^ω(x) = \sum_{ \mathbf{i}\in \mathbb{Z}^d} q_{\mathbf{i}}(ω) W(x-\mathbf{i})$ is a random nonnegative alloy-type potential with compactly supported single site potential $W$. We prove that there are constants $C, \widetilde C,D, \widetilde D>0$ such that $$ -C \leq\liminf_{λ\searrow 0} \frac{λ^{d/α}}{|\log F_q(D λ)|}{\log \ell(λ)} \qquad \text{and} \qquad \limsup_{λ\searrow 0} \frac{λ^{d/α}}{|\log F_q(\widetilde D λ)|}\log \ell(λ) \leq -\widetilde C, $$ where $F_q$ is the common cumulative distribution function of the lattice random variables $q_{\mathbf i}$. In particular, we identify how the behavior of $\ell$ at zero depends on the lattice configuration. For typical examples of $F_q$ the constants $D$ and $\widetilde D$ can be eliminated from the statement above. We combine probabilistic and analytic methods which allow to treat, in a unified manner, both local and non-local kinetic terms such as the Laplace operator, its fractional powers and the quasi-relativistic Hamiltonians.

math.PR↗
arXiv2019

Lifschitz tail for alloy-type models driven by the fractional Laplacian

We establish precise asymptotics near zero of the integrated density of states for the random Schrödinger operators $(-Δ)^{α/2} + V^ω$ in $L^2(\mathbb R^d)$ for the full range of $α\in(0,2]$ and a fairly large class of random nonnegative alloy-type potentials $V^ω$. The IDS exhibits the Lifschitz tail singularity. We prove the existence of the limit $$\lim_{s\to 0} s^{d/α}\ln\ell([0,s]) = -C \left(λ_d^{(α)}\right)^{d/α},$$ with $C \in (0,\infty]$. The constant $C$ is is finite if and only if the common distribution of the lattice random variables charges $\left\{0\right\}$. In this case, the constant $C$ is expressed explicitly in terms of such a probability. In the limit formula, $λ_d^{(α)}$ denotes the Dirichlet ground-state eigenvalue of the operator $(-Δ)^{α/2}$ in the unit ball in $\mathbb R^d.$

math.PR↗
arXiv2019

Progressive intrinsic ultracontractivity and heat kernel estimates for non-local Schrödinger operators

We study the long-time asymptotic behaviour of semigroups generated by non-local Schrödinger operators of the form $H = -L+V$; the free operator $L$ is the generator of a symmetric Lévy process in $\mathbb R^d$, $d > 1$ (with non-degenerate jump measure) and $V$ is a sufficiently regular confining potential. We establish sharp two-sided estimates of the corresponding heat kernels for large times and identify a new general regularity property, which we call progressive intrinsic ultracontractivity, to describe the large-time evolution of the corresponding Schrödinger semigroup. We discuss various examples and applications of these estimates, for instance we characterize the heat trace and heat content. Our examples cover a wide range of processes and we have to assume only mild restrictions on the growth, resp.\ decay, of the potential and the jump intensity of the free process. Our approach is based on a combination of probabilistic and analytic methods; our examples include fractional and quasi-relativistic Schrödinger operators.

math.PR↗