arXiv subjects
Krzysztof Bogdan
Publications and source records attributed to Krzysztof Bogdan.
Burkholder inequality by Bregman divergence
We prove Burkholder inequality using Bregman divergence.
Yaglom limit for unimodal Lévy processes
We prove universality of the Yaglom limit of Lipschitz cones among all unimodal Lévy processes sufficiently close to the isotropic $α$-stable Lévy process.
Optimal Hardy inequality for the fractional Laplacian on $L^p$
For the fractional Laplacian we give Hardy inequality which is optimal in $L^p$ for $1<p<\infty$. As an application, we explicitly characterize the contractivity of the corresponding Feynman-Kac semigroups on $L^p$.
Maximum likelihood estimation for discrete exponential families and random graphs
We characterize the existence of the maximum likelihood estimator for discrete exponential families. Our criterion is simple to apply as we show in various settings, most notably for exponential models of random graphs. As an application, we point out the size of independent identically distributed samples for which the maximum likelihood estimator exists with high probability.
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.
Sharp Gaussian estimates for heat kernels of Schrödinger operators
We characterize functions $V\le 0$ for which the heat kernel of the Schrö\-dinger operator $Δ+V$ is comparable with the Gauss-Weierstrass kernel uniformly in space and time. In dimension $4$ and higher the condition turns out to be more restrictive than the condition of the boundedness of the Newtonian potential of $V$. This resolves the question of V.~Liskevich and Y.~Semenov posed in 1998. We also give specialized sufficient conditions for the comparability, showing that local $L^p$ integrability of $V$ for $p>1$ is not necessary for the comparability.
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.
Heat kernel of anisotropic nonlocal operators
We construct and estimate the fundamental solution of highly anisotropic space-inhomogeneous integro-differential operators. We use the Levi method. We give applications to the Cauchy problem for such operators.
Yaglom limit for stable processes in cones
We give the asymptotics of the tail of the distribution of the first exit time of the isotropic $α$-stable Lévy process from the Lipschitz cone in $\mathbb{R}^d$. We obtain the Yaglom limit for the killed stable process for the cone. We construct and estimate entrance laws for the process from the vertex into the cone. For the symmetric Cauchy process and the positive half-line we give a spectral representation of the Yaglom limit. 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.
Sharp Gaussian estimates for Schrödinger heat kernels: $L^p$ integrability conditions
We give new sufficient conditions for comparability of the fundamental solution of the Schrödinger equation $\partial_t=Δ+V$ with the Gauss-Weierstrass kernel and show that local $L^p$ integrability of $V$ for $p> 1$ is not necessary for the comparability.
Characterization of sharp global Gaussian estimates for Schrödinger heat kernels
We investigate when the fundamental solution of the Schrödinger equation $\partial_t=Δ+V$ posseses sharp Gaussian bounds global in space and time. We give a characterization for $V\leq 0$ and a sufficient condition for general $V$.
Lévy systems and moment formulas for mixed Poisson integrals
We propose Mecke-Palm formulas for multiple integrals with respect to a Poisson random measure interlaced with its intensity measure. We apply such formulas to multiple mixed Lévy systems of Lévy processes and obtain moment formulas for products of interlaced multiple Poisson integrals.
Majorization, 4G Theorem and Schrödinger perturbations
Schrödinger perturbations of transition densities by singular potentials may fail to be comparable with the original transition density. For instance this is so for the transition density of a subordinator perturbed by any time-independent unbounded potential. In order to estimate such perturbations it is convenient to use an auxilary transition density as a majorant and the 4G inequality for the original transition density and the majorant. We prove the 4G inequality for the $1/2$-stable and inverse Gaussian subordinators, discuss the corresponding class of admissible potentials and indicate estimates for the resulting transition densities of Schrödinger operators. The connection of the transition densities to their generators is made via the weak-type notion of fundamental solution.
Hardy-Stein identities and square functions for semigroups
We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump Lévy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1<p<\infty$, of the corresponding Littlewood-Paley square function. The square function yields a direct proof of the $L^p$ boundedness of Fourier multipliers obtained by transforms of martingales of Lévy processes.
Trace estimates for unimodal Lévy processes
We give two-term small-time approximation for the trace of the Dirichlet heat kernel of bounded smooth domain for unimodal Lévy processes satisfying the weak scaling conditions.
Hardy inequalities and non-explosion results for semigroups
We prove non-explosion results for Schrödinger perturbations of symmetric transition densities and Hardy inequalities for their quadratic forms by using explicit supermedian functions of their semigroups.
Gaussian estimates for Schroedinger perturbations
We prove an optimal 4G Theorem for the Gaussian kernel. We also propose a new general method of estimating Schroedinger perturbations of transition densities, and give applications to the Gaussian kernel.