Tomasz Grzywny
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Research Subjects
Tomasz Grzywny has a total of 35 co-authors. Here are the top co-authors by number of shared publications:
- Krzysztof Bogdan
Department of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wrocław, Poland
10 shared publications - Karol Szczypkowski
Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology 2 , Wyb. Wyspiańskiego 27, 50-370 Wrocław
8 shared publications - Wojciech Cygan
Faculty of Mathematics and Computer Science University of Wroc law Institute of Math- ematics pl. Grunwaldzki 2/4 50-384 Wroc law Poland
8 shared publications - Julia Lenczewska
Wrocław University of Science and Technology, Faculty of Pure and Applied Mathematics, Wrocław, Poland
5 shared publications - Michał Ryznar
Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wrocław, Poland
5 shared publications - Łukasz Leżaj
Faculty of Pure and Applied Mathematics Wrocław University of Science and Technology Wyb. Wyspiańskiego 27 50-370 Wrocław POLAND
5 shared publications
Latest Publications
- 23 January 2026
- Published by Springer Nature in Fractional Calculus and Applied Analysis
- Vol. 29 (1) , 494-542
Asymptotics and geometric flows for a class of nonlocal curvatures
- 23 January 2026
- Published by Springer Nature in Fractional Calculus and Applied Analysis
- Vol. 29 (1) , 494-542
Asymptotics and geometric flows for a class of nonlocal curvatures
- 12 August 2025
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
Hardy inequalities and nonlocal capacity
In this article, we introduce and study capacities related to nonlocal Sobolev spaces, with focus on spaces corresponding to zero-order nonlocal operators. In particular, we prove Hardy-type inequalities to obtain Sobolev embeddings and use them to estimate the nonlocal capacities of a ball.
- 12 August 2025
- Published by American Mathematical Society (AMS) in Proceedings of the American Mathematical Society
Hardy inequalities and nonlocal capacity
In this article, we introduce and study capacities related to nonlocal Sobolev spaces, with focus on spaces corresponding to zero-order nonlocal operators. In particular, we prove Hardy-type inequalities to obtain Sobolev embeddings and use them to estimate the nonlocal capacities of a ball.
- 1 April 2025
- Published by AIP Publishing in Chaos: An Interdisciplinary Journal of Nonlinear Science
- Vol. 35 (4)
Partial versus total resetting for Lévy flights in d dimensions: Similarities and discrepancies
While stochastic resetting (or total resetting) is a less young and more established concept in stochastic processes, partial stochastic resetting (PSR) is a relatively new field. PSR means that, at random moments in time, a stochastic process gets multiplied by a factor between 0 and 1, thus approaching but not reaching the resetting position. In this paper, we present new results on PSR highlighting the main similarities and discrepancies with total resetting. Specifically, we consider both symmetric α-stable Lévy processes (Lévy flights) and Brownian motion with PSR in arbitrary d dimensions. We derive explicit expressions for the propagator and its stationary measure and discuss in detail their asymptotic behavior. Interestingly, while approaching to stationarity, a dynamical phase transition occurs for the Brownian motion, but not for Lévy flights. We also analyze the behavior of the process around the resetting position and find significant differences between PSR and total resetting.
- 1 April 2025
- Published by AIP Publishing in Chaos: An Interdisciplinary Journal of Nonlinear Science
- Vol. 35 (4)
Partial versus total resetting for Lévy flights in d dimensions: Similarities and discrepancies
While stochastic resetting (or total resetting) is a less young and more established concept in stochastic processes, partial stochastic resetting (PSR) is a relatively new field. PSR means that, at random moments in time, a stochastic process gets multiplied by a factor between 0 and 1, thus approaching but not reaching the resetting position. In this paper, we present new results on PSR highlighting the main similarities and discrepancies with total resetting. Specifically, we consider both symmetric α-stable Lévy processes (Lévy flights) and Brownian motion with PSR in arbitrary d dimensions. We derive explicit expressions for the propagator and its stationary measure and discuss in detail their asymptotic behavior. Interestingly, while approaching to stationarity, a dynamical phase transition occurs for the Brownian motion, but not for Lévy flights. We also analyze the behavior of the process around the resetting position and find significant differences between PSR and total resetting.
- 24 December 2024
- Published by Springer Nature in Mathematische Annalen
- Vol. 391 (4) , 5857-5910
Liouville’s theorems for Lévy operators
- 24 December 2024
- Published by Springer Nature in Mathematische Annalen
- Vol. 391 (4) , 5857-5910
Liouville’s theorems for Lévy operators
- Open Access
Stationary states for stable processes with partial resetting
We study a $d$-dimensional stochastic process $\mathbf{X}$ which arises from a Lévy process $\mathbf{Y}$ by partial resetting, that is the position of the process $\mathbf{X}$ at a Poisson moment equals $c$ times its position right before the moment, and it develops as $\mathbf{Y}$ between these two consecutive moments, $c \in (0, 1)$. We focus on $\mathbf{Y}$ being a strictly $α$-stable process with $α\in (0,2]$ having a transition density: We analyze properties of the transition density $p$ of the process $\mathbf{X}$. We establish a series representation of $p$. We prove its convergence as time goes to infinity (ergodicity), and we show that the limit $ρ_{\mathbf{Y}}$ (density of the ergodic measure) can be expressed by means of the transition density of the process $\mathbf{Y}$ starting from zero, which results in closed concise formulae for its moments. We show that the process $\mathbf{X}$ reaches a non-equilibrium stationary state. Furthermore, we check that $p$ satisfies the Fokker--Planck equation, and we confirm the harmonicity of $ρ_{\mathbf{Y}}$ with respect to the adjoint generator. In detail, we discuss the following cases: Brownian motion, isotropic and $d$-cylindrical $α$-stable processes for $α\in (0,2)$, and $α$-stable subordinator for $α\in (0,1)$. We find the asymptotic behavior of $p(t;x,y)$ as $t\to +\infty$ while $(t,y)$ stays in a certain space-time region. For Brownian motion, we discover a phase transition, that is a change of the asymptotic behavior of $p(t;0,y)$ with respect to $ρ_{\mathbf{Y}}(y)$.Open AccessStationary states for stable processes with partial resetting
We study a $d$-dimensional stochastic process $\mathbf{X}$ which arises from a Lévy process $\mathbf{Y}$ by partial resetting, that is the position of the process $\mathbf{X}$ at a Poisson moment equals $c$ times its position right before the moment, and it develops as $\mathbf{Y}$ between these two consecutive moments, $c \in (0, 1)$. We focus on $\mathbf{Y}$ being a strictly $α$-stable process with $α\in (0,2]$ having a transition density: We analyze properties of the transition density $p$ of the process $\mathbf{X}$. We establish a series representation of $p$. We prove its convergence as time goes to infinity (ergodicity), and we show that the limit $ρ_{\mathbf{Y}}$ (density of the ergodic measure) can be expressed by means of the transition density of the process $\mathbf{Y}$ starting from zero, which results in closed concise formulae for its moments. We show that the process $\mathbf{X}$ reaches a non-equilibrium stationary state. Furthermore, we check that $p$ satisfies the Fokker--Planck equation, and we confirm the harmonicity of $ρ_{\mathbf{Y}}$ with respect to the adjoint generator. In detail, we discuss the following cases: Brownian motion, isotropic and $d$-cylindrical $α$-stable processes for $α\in (0,2)$, and $α$-stable subordinator for $α\in (0,1)$. We find the asymptotic behavior of $p(t;x,y)$ as $t\to +\infty$ while $(t,y)$ stays in a certain space-time region. For Brownian motion, we discover a phase transition, that is a change of the asymptotic behavior of $p(t;0,y)$ with respect to $ρ_{\mathbf{Y}}(y)$. - Open Access
Alexandrov Theorem for nonlocal curvature
In this article we obtain a nonlocal version of the Alexandrov Theorem which asserts that the only set with sufficiently smooth boundary and of constant nonlocal mean curvature is an Euclidean ball. We consider a general nonlocal mean curvature given by a radial and monotone kernel and we formulate an easy-to-check condition which is necessary and sufficient for the nonlocal version of the Alexandrov Theorem to hold in the treated context. Our definition encompasses numerous examples of various nonlocal mean curvatures that have been already studied in the literature. To prove the main result we obtain a specific formula for the tangential derivative of the nonlocal mean curvature and combine it with an application of the method of moving planes.
Open AccessAlexandrov Theorem for nonlocal curvature
In this article we obtain a nonlocal version of the Alexandrov Theorem which asserts that the only set with sufficiently smooth boundary and of constant nonlocal mean curvature is an Euclidean ball. We consider a general nonlocal mean curvature given by a radial and monotone kernel and we formulate an easy-to-check condition which is necessary and sufficient for the nonlocal version of the Alexandrov Theorem to hold in the treated context. Our definition encompasses numerous examples of various nonlocal mean curvatures that have been already studied in the literature. To prove the main result we obtain a specific formula for the tangential derivative of the nonlocal mean curvature and combine it with an application of the method of moving planes.
- Open Access
Hardy inequalities and nonlocal capacity
In this article, we introduce and study capacities related to nonlocal Sobolev spaces, with focus on spaces corresponding to zero-order nonlocal operators. In particular, we prove Hardy-type inequalities to obtain Sobolev embeddings and use them to estimate the nonlocal capacities of a ball.
Open AccessHardy inequalities and nonlocal capacity
In this article, we introduce and study capacities related to nonlocal Sobolev spaces, with focus on spaces corresponding to zero-order nonlocal operators. In particular, we prove Hardy-type inequalities to obtain Sobolev embeddings and use them to estimate the nonlocal capacities of a ball.
- Open Access
- 15 May 2023
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 62 (5) , 1-31
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. As an application, we obtain the boundedness of the Dirichlet-to-Neumann operator on weighted $$L^p$$ spaces. We also show that the Poisson integrals are quasiminimizers of the Sobolev–Bregman forms.- 15 May 2023
- Published by Springer Nature in Calculus of Variations and Partial Differential Equations
- Vol. 62 (5) , 1-31
Open AccessNonlinear 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. As an application, we obtain the boundedness of the Dirichlet-to-Neumann operator on weighted $$L^p$$ spaces. We also show that the Poisson integrals are quasiminimizers of the Sobolev–Bregman forms. - Open Access
- 24 April 2023
- Published by Springer Nature in Annali di Matematica Pura ed Applicata (1923 -)
- Vol. 202 (6) , 2629-2651
Asymptotics of non-local perimeters
We introduce a notion of non-local perimeter which is defined through an arbitrary positive Borel measure on $${\mathbb {R}}^d$$ which integrates the function $$1\wedge |x|$$ . Such definition of non-local perimeter encompasses a wide range of perimeters which have been already studied in the literature, including fractional perimeters and anisotropic fractional perimeters. The main part of the article is devoted to the study of the asymptotic behaviour of non-local perimeters. As direct applications we recover well-known convergence results for fractional perimeters and anisotropic fractional perimeters.- 24 April 2023
- Published by Springer Nature in Annali di Matematica Pura ed Applicata (1923 -)
- Vol. 202 (6) , 2629-2651
Open AccessAsymptotics of non-local perimeters
We introduce a notion of non-local perimeter which is defined through an arbitrary positive Borel measure on $${\mathbb {R}}^d$$ which integrates the function $$1\wedge |x|$$ . Such definition of non-local perimeter encompasses a wide range of perimeters which have been already studied in the literature, including fractional perimeters and anisotropic fractional perimeters. The main part of the article is devoted to the study of the asymptotic behaviour of non-local perimeters. As direct applications we recover well-known convergence results for fractional perimeters and anisotropic fractional perimeters. - Open Access
Asymptotic expansion of the nonlocal heat content
Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a L\'evy process in $\mathbb{R}^d$ and $\Omega$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H(t)=\int_{\Omega} \mathbb{P}^x (X_t\in\Omega^c) \, \mathrm{d}x$ which is called the heat content. We study its asymptotic expansion for isotropic $\alpha$-stable L\'evy processes and more general L\'evy processes, under mild assumptions on the characteristic exponent.Open AccessAsymptotic expansion of the nonlocal heat content
Let $\mathbf{X}=\{X_t\}_{t\geq 0}$ be a L\'evy process in $\mathbb{R}^d$ and $\Omega$ be an open subset of $\mathbb{R}^d$ with finite Lebesgue measure. In this article we consider the quantity $H(t)=\int_{\Omega} \mathbb{P}^x (X_t\in\Omega^c) \, \mathrm{d}x$ which is called the heat content. We study its asymptotic expansion for isotropic $\alpha$-stable L\'evy processes and more general L\'evy processes, under mild assumptions on the characteristic exponent.