Bregman variation of semimartingales
We define Bregman variation of semimartingales. We give its pathwise representation, Itô-type isometry for martingales, and applications to harmonic analysis.
math.PR↗
arXiv subjects
Krzysztof Bogdan
Publications and source records attributed to Krzysztof Bogdan.
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↗
Critical fractional Hardy inequalities
We give a roadmap for the study of critical fractional Hardy inequalities for Dirichlet and Sobolev--Bregman forms. We focus on homogeneous forms on the real line, in particular the form of the fractional Laplacian and the Servadei--Valdinoci form. We prove criticality results in both the Dirichlet and Sobolev--Bregman settings.
math.AP↗
Heat kernel bounds for the fractional Laplacian with Hardy potential in angular momentum channels
Motivated by the study of relativistic atoms, we prove sharp heat kernel bounds for the Hardy operator $(-Δ)^{α/2}-κ|x|^{-α}$ acting on functions of the form $u(|x|) |x|^{\ell} Y_{\ell,m}(x/|x|)$ in $L^2(\R^d)$, when $α\in(0,2]\cap(0,d+2\ell)$.
math.AP↗
Subordinated Bessel heat kernels
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.
math.AP↗
Polarized Hardy--Stein identity
We prove the Hardy--Stein identity for vector functions in $L^p(\mathbb R^d;\mathbb R^n)$ with $1<p<\infty$ and for the canonical paring of two real functions in $L^p(\mathbb R^d)$ with $2\le p<\infty$. To this end we propose a notion of Bregman co-divergence and study the corresponding integral forms.
math.CA↗
Shot-down stable processes
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 $D$.
math.PR↗
Nonlocal operators with Neumann conditions
We construct a strong Markov process corresponding to the Dirichlet form of Servadei and Valdinoci and use the process to solve the corresponding Neumann boundary problem for the fractional Laplacian and the half-line.
math.PR↗
Stable processes with reflection
We construct a Hunt process that can be described as an isotropic $α$-stable Lévy 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 Schrödinger 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.
math.PR↗
Hardy perturbations of subordinated Bessel heat kernels
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.
math.AP↗
Caloric functions and boundary regularity for the fractional Laplacian in Lipschitz open sets
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 $α$-stable Lévy 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 Hölder continuous up to the boundary.
math.AP↗
Self-similar solution for fractional Laplacian in cones
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 Lévy process and precise large-time asymptotics for solutions of the Cauchy problem in the cone.
math.PR↗
The fractional Laplacian with reflections
Motivated by the notion of isotropic $α$-stable Lévy 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.
math.PR↗
Ground state representation for the fractional Laplacian with Hardy potential in angular momentum channels
Motivated by the study of relativistic atoms, we consider the Hardy operator $(-Δ)^{α/2}-κ|x|^{-α}$ acting on functions of the form $u(|x|) |x|^{\ell} Y_{\ell,m}(x/|x|)$ in $L^2(\mathbb{R}^d)$, when $κ\geq0$ and $α\in(0,2]\cap(0,d+2\ell)$. 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.
math.AP↗
Self-similar solution for Hardy operator
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.
math.AP↗
Positive harmonically bounded solutions for semi-linear equations
For open sets $U$ in some space $X$, we are interested in positive solutions to semi-linear equations $ Lu=φ(\cdot,u)μ$ 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), $μ$ is a positive measure on $U$ and $φ$ is an arbitrary measurable real function on $U\times \mathbb{R}^+$ such that the functions $t\mapsto φ(x,t)$, $x\in U$, 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 $u+Kφ(\cdot,u)=g$ 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.
math.PR↗
Nonlinear nonlocal Douglas identity
We give Hardy-Stein and Douglas identities for nonlinear nonlocal Sobolev-Bregman integral forms with unimodal Lévy 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.
math.AP↗
Anomalous diffusion limit for a kinetic equation with a thermostatted interface
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.
math.PR↗