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Tomasz Grzywny

Publications and source records attributed to Tomasz Grzywny.

At least 37 records · Page 2Linked to original sources
arXiv2019

L{é}vy processes: concentration function and heat kernel bounds

We investigate densities of vaguely continuous convolution semigroups of probability measures on $\mathbb{R}^d$. We expose that many typical conditions on the characteristic exponent repeatedly used in the literature of the subject are equivalent to the behaviour of the maximum of the density as a function of time variable. We also prove qualitative lower estimates under mild assumptions on the corresponding jump measure and the characteristic exponent.

math.PR↗
arXiv2019

Estimates of Dirichlet heat kernel for symmetric Markov processes

We consider a large class of symmetric pure jump Markov processes dominated by isotropic unimodal Lévy processes with weak scaling conditions. First, we establish sharp two-sided heat kernel estimates for these processes in $C^{1,1}$ open sets. As corollaries of our main results, we obtain sharp two-sided Green function estimates and a scale invariant boundary Harnack inequality with explicit decay rates in $C^{1,1}$ open sets.

math.PR↗
arXiv2018

Extension and trace for nonlocal operators

We prove an optimal extension and trace theorem for Sobolev spaces of nonlocal operators. The extension is given by a suitable Poisson integral and solves the corresponding nonlocal Dirichlet problem. We give a Douglas-type formula for the quadratic form of the Poisson extension.

math.AP↗
arXiv2018

Remarks on the nonlocal Dirichlet problem

We study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.

math.AP↗
arXiv2018

Heat kernels of non-symmetric Lévy-type operators

We construct the fundamental solution (the heat kernel) $p^κ$ to the equation $\partial_t=\mathcal{L}^κ$, where under certain assumptions the operator $\mathcal{L}^κ$ takes one of the following forms, \begin{align*} \mathcal{L}^κf(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left )κ(x,z)J(z)\, dz \,, \mathcal{L}^κf(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x))κ(x,z)J(z)\, dz\,, \mathcal{L}^κf(x)&:= \frac1{2}\int_{\mathbb{R}^d}( f(x+z)+f(x-z)-2f(x))κ(x,z)J(z)\, dz\,. \end{align*} In particular, $J\colon \mathbb{R}^d \to [0,\infty]$ is a Lévy density, i.e., $\int_{\mathbb{R}^d}(1\land |x|^2)J(x)dx<\infty$. The function $κ(x,z)$ is assumed to be Borel measurable on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<κ_0\leq κ(x,z)\leq κ_1$, and $|κ(x,z)-κ(y,z)|\leq κ_2|x-y|^β$ for some $β\in (0, 1)$. We prove the uniqueness, estimates, regularity and other qualitative properties of $p^κ$.

math.AP↗
arXiv2018

A note on the generalized heat content for Lévy processes

Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a Lévy process in $\mathbb{R}^d$ and $Ω$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. The quantity $H (t) = \int_Ω \mathbb{P}^{x} (X_t\in Ω^c) d x$ is called the heat content. In this article we consider its generalized version $H_g^μ(t) = \int_{\mathbb{R}^d}\mathbb{E}^{x} g(X_t)μ( d x )$, where $g$ is a bounded function and $μ$ a finite Borel measure. We study its asymptotic behaviour at zero for various classes of Lévy processes.

math.PR↗
arXiv2017

Fractional Laplacian with Hardy potential

We give sharp two-sided estimates of the semigroup generated by the fractional Laplacian plus the Hardy potential on $\mathbb{R}^d$, including the case of the critical constant. We use Davies' method back-to-back with a new method of integral analysis of Duhamel's formula.

math.AP↗
arXiv2017

Spectral Heat Content for Lévy Processes

In this paper we study the spectral heat content for various Lévy processes. We establish the asymptotic behavior of the spectral heat content for Lévy processes of bounded variation in $\mathbb{R}^{d}$, $d\geq 1$. We also study the spectral heat content for arbitrary open sets of finite Lebesgue measure in $\mathbb{R}$ with respect to Lévy processes of unbounded variation under certain conditions on their characteristic exponents. Finally we establish that the asymptotic behavior of the spectral heat content is stable under integrable perturbations to the Lévy measure.

math.PR↗
arXiv2017

Potential kernels, probabilities of hitting a ball, harmonic functions and the boundary Harnack inequality for unimodal Lévy processes

In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal Lévy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the Lévy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to $O$-regular variation at zero and at infinity with lower indices greater than $-d - 2$. We also prove a supremum estimate and a regularity result for functions harmonic with respect to a general isotropic unimodal Lévy process. In the second part we apply the recent results on the boundary Harnack inequality and Martin representation of harmonic functions for the class of isotropic unimodal Lévy processes characterised by a localised version of the scaling-type condition mentioned above. As a sample application, we provide sharp two-sided estimates of the Green function of a half-space. Our results are expressed in terms of Pruitt's functions $K(r)$ and $L(r)$, measuring local activity and the amount of large jumps of the Lévy process, respectively.

math.PR↗
arXiv2016

Heat content for convolution semigroups

Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a Lévy process in $\mathbb{R}^d$ and $Ω$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H (t) = \int_Ω\mathbb{P}_{x} (X_t\in Ω^c) dx$ which is called the heat content. We study its asymptotic behaviour as $t$ goes to zero for isotropic Lévy processes under some mild assumptions on the characteristic exponent. We also treat the class of Lévy processes with finite variation in full generality.

math.PR↗
arXiv2016

Kato classes for Lévy processes

We prove that the definitions of the Kato class by the semigroup and by the resolvent of the Lévy process on $\mathbb{R}^d$ coincide if and only if 0 is not regular for {0}. If 0 is regular for {0} then we describe both classes in detail. We also give an analytic reformulation of these results by means of the characteristic (Lévy-Khintchine) exponent of the process. The result applies to the time-dependent (non-autonomous) Kato class. As one of the consequences we obtain a simultaneous time-space smallness condition equivalent to the Kato class condition given by the semigroup.

math.FA↗
arXiv2016

Asymptotic behaviour and estimates of slowly varying convolution semigroups

We prove the asymptotic formulas for the transition densities of isotropic unimodal convolution semigroups of probability measures on $\mathbb{R} ^d$ under the assumption that its Lévy--Khintchine exponent varies slowly. We also derive some new estimates of the transition densities and Green functions.

math.PR↗
arXiv2015

Green function for gradient perturbation of unimodal Lévy processes

We prove that the Green function of a generator of isotropic unimodal Lévy processes with the weak lower scaling order bigger than one and the Green function of its gradient perturbations are comparable for bounded smooth open sets if the drift function is from an appropriate Kato class.

math.AP↗
arXiv2015

Hitting times of points and intervals for symmetric Lévy processes

For one-dimensional symmetric Lévy processes, which hit every point with positive probability, we give sharp bounds for the tail function of the first hitting time of B which is either a single point or an interval. The estimates are obtained under some weak type scaling assumptions on the characteristic exponent of the process. We apply these results to prove optimal estimates of the transition density of the process killed after hitting B.

math.PR↗
arXiv2013

On Harnack inequality and Hölder regularity for isotropic unimodal Lévy processes

We prove the scale invariant Harnack inequality and regularity properties for harmonic functions with respect to an isotropic unimodal Lévy process with the characteristic exponent $ψ$ satisfying some scaling condition. We show sharp estimates of the potential measure and capacity of balls, and further, under the assumption of that $ψ$ satisfies the lower scaling condition, sharp estimates of the potential kernel of the underlying process. This allow us to establish the Krylov-Safonov type estimate, which is the key ingredient in the approach of Bass and Levin, that we follow.

math.PR↗