Search

We construct a Hunt process that can be described as an isotropic $\alpha$-stable L\'evy process reflected from the complement of a bounded open Lipschitz set. In fact, we introduce a new analytic method for concatenating Markov processes. It is based on nonlocal Schr\"odinger perturbations of sub-Markovian transition kernels and the construction of two supermedian functions with different growth rates at infinity. We apply this framework to describe the return distribution and the stationary distribution of the process. To handle the strong Markov property at the reflection time, we introduce a novel ladder process, whose transition semigroup encodes not only the position of the process, but also the number of reflections., Comment: 21 pages, no figures

Motivated by the spectral theory of relativistic atoms, we prove matching upper and lower bounds for the transition density of Hardy perturbations of subordinated Bessel heat kernels. The analysis is based on suitable supermedian functions, in particular invariant functions., Comment: 59 pages

We give Martin representation of nonnegative functions caloric with respect to the fractional Laplacian in Lipschitz open sets. The caloric functions are defined in terms of the mean value property for the space-time isotropic $\alpha$-stable L\'evy process. To derive the representation, we first establish the existence of the parabolic Martin kernel. This involves proving new boundary regularity results for both the fractional heat equation and the fractional Poisson equation with Dirichlet exterior conditions. Specifically, we demonstrate that the ratio of the solution and the Green function is H\"older continuous up to the boundary., Comment: 40 pages

We prove the Hardy--Stein identity for vector functions in $L^p(\mathbb R^d;\mathbb R^n)$ with $1

We prove new bounds for Bessel heat kernels and Bessel heat kernels subordinated by stable subordinators. In particular, we provide a 3G inequality in the subordinated case., Comment: 18 pages

We construct a self-similar solution of the heat equation for the fractional Laplacian with Dirichlet boundary conditions in every fat cone. As applications, we give the Yaglom limit and entrance law for the corresponding killed isotropic stable L\'{e}vy process and precise large-time asymptotics for solutions of the Cauchy problem in the cone., Comment: We provided some examples to illustrate our findings

Motivated by the study of relativistic atoms, we consider the Hardy operator $(-\Delta)^{\alpha/2}-\kappa|x|^{-\alpha}$ acting on functions of the form $u(|x|) |x|^{\ell} Y_{\ell,m}(x/|x|)$ in $L^2(\mathbb{R}^d)$, when $\kappa\geq0$ and $\alpha\in(0,2]\cap(0,d+2\ell)$. We give a ground state representation of the corresponding form on the half-line (Theorem 1.5). For the proof we use subordinated Bessel heat kernels., Comment: 32 pages

The shot-down process is a strong Markov process which is annihilated, or shot down, when jumping over or to the complement of a given open subset of a vector space. Due to specific features of the shot-down time, such processes suggest new type of boundary conditions for nonlocal differential equations. In this work we construct the shot-down process for the fractional Laplacian in Euclidean space. For smooth bounded sets $D$, we study its transition density and characterize Dirichlet form. We show that the corresponding Green function is comparable to that of the fractional Laplacian with Dirichlet conditions on $D$. However, for nonconvex $D$, the transition density of the shot-down stable process is incomparable with the Dirichlet heat kernel of the fractional Laplacian for $D$. Furthermore, Harnack inequality in general fails for harmonic functions of the shot-down process., Comment: 39 pages

For open sets $U$ in some space $X$, we are interested in positive solutions to semi-linear equations $ Lu=\varphi(\cdot,u)\mu$ on $U$. Here $L$ may be an elliptic or parabolic operator of second order (generator of a diffusion process) or an integro-differential operator (generator of a jump process), $\mu$ is a positive measure on $U$ and $\varphi$ is an arbitrary measurable real function on $U\times \mathbb{R}^+$ such that the functions $t\mapsto \varphi(x,t)$, $x\in U$, are continuous, increasing and vanish at $t=0$. More precisely, given a measurable function $h\ge 0$ on $X$ which is $L$-harmonic on $U$, that is, continuous real on $U$ with $Lh=0$ on $U$, we give necessary and sufficient conditions for the existence of positive solutions $u$ such that $u=h$ on $X\setminus U$ and $u$ has the same ``boundary behavior'' as $h$ on $U$ (Problem 1) or, alternatively, $u\le h$ on $U$, but $u\not\equiv 0$ on $U$ (Problem 2). We show that these problems are equivalent to problems of the existence of positive solutions to certain integral equations $u+K\varphi(\cdot,u)=g$ on $U$, $K$ being a potential kernel. We solve them in the general setting of balayage spaces $(X,\mathcal{W})$ which, in probabilistic terms, corresponds to the setting of transient Hunt processes with strong Feller resolvent., Comment: 39 pages, updated authors' details

Motivated by the notion of isotropic $\alpha$-stable L\'evy processes confined, by reflections, to a bounded open Lipschitz set $D\subset \mathbb{R}^d$, we study some related analytical objects. Thus, we construct the corresponding transition semigroup, identify its generator and prove exponential speed of convergence of the semigroup to a unique stationary distribution for large time., Comment: 30 pages, no figures. Revision based on referee's comments

We consider the limit of solutions of scaled linear kinetic equations with a reflection-transmission-absorption condition at the interface. Both the coefficient describing the probability of absorption and the scattering kernel degenerate. We prove that the long-time, large-space limit is the unique solution of a version of the fractional in space heat equation that corresponds to the Kolmogorov equation for a symmetric stable process, which is reflected, or transmitted while crossing the interface and is killed upon the first hitting of the interface. The results of the paper are related to the work in [KOR20], where the case of a non-degenerate probability of absorption has been considered.

We prove a Douglas-type identity in $L^p$ for $1

We describe the large-time asymptotics of solutions to the heat equation for the fractional Laplacian with added subcritical or even critical Hardy-type potential. The asymptotics is governed by a self-similar solution of the equation, obtained as a normalized limit at the origin of the kernel of the corresponding Feynman-Kac semigroup., Comment: 27 pages, corrected exponent in Theorem 4.6

We prove universality of the Yaglom limit of Lipschitz cones among all unimodal L\'{e}vy processes sufficiently close to the isotropic $\alpha$-stable L\'{e}vy process., Comment: 36 pages, 1 figure

For the fractional Laplacian we give Hardy inequality which is optimal in $L^p$ for $1

We prove Burkholder inequality using Bregman divergence., Comment: We updated a reference, corrected constants c_p and C_p, and made small editorial changes. The paper will appear in the Bulletin of the Polish Academy of Sciences. Mathematics

We give Hardy-Stein and Douglas identities for nonlinear nonlocal Sobolev-Bregman integral forms with unimodal L\'evy measures. We prove that the corresponding Poisson integral defines an extension operator for the Sobolev-Bregman spaces. We also show that the Poisson integrals are quasiminimizers of the Sobolev-Bregman forms., Comment: 29 pages; we added applications to the Dirichlet-to-Neumann map, in Section 6

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., Comment: 21 pages, minor editorial changes, added connections to the criterion of Barndorff-Nielsen and linear programming

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., Comment: small editorial changes, added literature, new title, 28 pages, 1 figure

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., Comment: We are now able to add the trace theorem and complete the picture

We characterize functions $V\le 0$ for which the heat kernel of the Schr\"o\-dinger operator $\Delta+V$ 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., Comment: 19 pages. Editorial changes. Theorem 1.4 has a shorter proof. The paper merges results of arXiv:1606.03745 and arXiv:1511.07167

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., Comment: 51 pages

We give the asymptotics of the tail of the distribution of the first exit time of the isotropic $\alpha$-stable L\'evy 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., Comment: 17 pages

We investigate when the fundamental solution of the Schr\"odinger equation $\partial_t=\Delta+V$ posseses sharp Gaussian bounds global in space and time. We give a characterization for $V\leq 0$ and a sufficient condition for general $V$.

We give new sufficient conditions for comparability of the fundamental solution of the Schr\"odinger equation $\partial_t=\Delta+V$ with the Gauss-Weierstrass kernel and show that local $L^p$ integrability of $V$ for $p> 1$ is not necessary for the comparability., Comment: 13 pages, added Example 3.5, small editorial changes

We prove a Hardy-Stein type identity for the semigroups of symmetric, pure-jump L\'evy processes. Combined with the Burkholder-Gundy inequalities, it gives the $L^p$ two-way boundedness, for $1

We give two-term small-time approximation for the trace of the Dirichlet heat kernel of bounded smooth domain for unimodal L\'evy processes satisfying the weak scaling conditions., Comment: 17 pages, 1 figure

We prove non-explosion results for Schr\"odinger perturbations of symmetric transition densities and Hardy inequalities for their quadratic forms by using explicit supermedian functions of their semigroups., Comment: 21 pages, updated references

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 L\'evy systems of L\'evy processes and obtain moment formulas for products of interlaced multiple Poisson integrals., Comment: added example on p. 24-25

Schr\"odinger 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 Schr\"odinger operators. The connection of the transition densities to their generators is made via the weak-type notion of fundamental solution., Comment: 17 pages, editorial changes and some details added in proofs, to appear in Journal of Evolution Equations

We give a power law for the homogeneity degree of the Martin kernel of the fractional Laplacian for the right circular cone when the angle of the cone tends to zero., Comment: 22 pages, 2 figures

We add a divergence-free drift with increasing magnitude to the fractional Laplacian on a bounded smooth domain, and discuss the behavior of the principal eigenvalue for the Dirichlet problem. The eigenvalue remains bounded if and only if the drift has non-trivial first integrals in the domain of the quadratic form of the fractional Laplacian., Comment: 19 pages

We construct superharmonic functions and give sharp bounds for the expected exit time and probability of survival for isotropic unimodal L\'evy processes, Comment: 33 pages, 1 figure, minor additions and editorial changes

We give sharp bounds for the isotropic unimodal probability convolution semigroups when their L\'evy-Khintchine exponent has Matuszewska indices strictly between 0 and 2., Comment: 23 pages, editorial changes, connection to Matuszewska indices explained, to appear in Journal of Functional Analysis

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., Comment: 24 pages, to appear in Studia Mathematica, minor editorial changes in this version

We prove a boundary Harnack inequality for jump-type Markov processes on metric measure state spaces, under comparability estimates of the jump kernel and Urysohn-type property of the domain of the generator of the process. The result holds for positive harmonic functions in arbitrary open sets. It applies, e.g., to many subordinate Brownian motions, L\'evy processes with and without continuous part, stable-like and censored stable processes, jump processes on fractals, and rather general Schr\"odinger, drift and jump perturbations of such processes., Comment: 37 pages, 1 figure, minor editorial changes, paper accepted in Transactions of AMS

We give a class of Fourier multipliers with non-symmetric symbols and explicit norm bounds on $L^p$ spaces by using the stochastic calculus of L\'evy processes and Burkholder-Wang estimates for differentially subordinate martingales., Comment: 14 pages

We give sufficient conditions for a nonlocal perturbation of an integral kernel to be locally in time comparable with the kernel., Comment: 15 pages, the perturbation may now be a general forward kernel, allowing for jumps and delays

For integral kernels on space-time we indicate a class of nonnegative Schr\"odinger perturbations which produce comparable integral kernels., Comment: 14 pages

We study iterations of integral kernels satisfying a transience-type condition and we prove exponential estimates analogous to Gronwall\rq{}s inequality. As a consequence we obtain estimates of Schr\"odinger perturbations of integral kernels, including Markovian semigroups., Comment: 17 pages, revised: Example 4.1 added

We characterize conditional Hardy spaces of the Laplacian and of the fractional Laplacian by using Hardy-Stein type identities., Comment: 23 pages, editorial changes and updated references, to appear in Hiroshima Math. J