Karol Szczypkowski
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Karol Szczypkowski has a total of 15 co-authors. Here are the top co-authors by number of shared publications:
- Tomasz Grzywny
Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wrocław, Poland
8 shared publications - Krzysztof Bogdan
Department of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wrocław, Poland
7 shared publications - Jacek Dziubański
Instytut Matematyczny, Uniwersytet Wrocławski, Wrocław, Poland
4 shared publications - Tomasz Jakubowski
Wrocław University of Science and Technology, Wrocław, POLAND
4 shared publications - Kamil Kaleta
Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, Wrocław, Poland
3 shared publications - Bartosz Trojan
Wydział Matematyki, Politechnika Wrocławska, Wrocław, Poland
2 shared publications
Latest Publications
- 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.
- 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
Non-symmetric Lévy-type operators
We present a general approach to the parametrix construction. We apply it to prove the uniqueness and existence of a weak fundamental solution for the equation $\partial_t =\mathcal{L}$ with non-symmetric non-local operators $$ \mathcal{L}f(x):= b(x)\cdot \nabla f(x)+ \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left)\kappa(x,z)J(z)\, dz\,, $$ under certain assumptions on $b$, $\kappa$ and $J$. The result allows more general coefficients even for $J(z)=|z|^{-d-1}$.Open AccessNon-symmetric Lévy-type operators
We present a general approach to the parametrix construction. We apply it to prove the uniqueness and existence of a weak fundamental solution for the equation $\partial_t =\mathcal{L}$ with non-symmetric non-local operators $$ \mathcal{L}f(x):= b(x)\cdot \nabla f(x)+ \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left)\kappa(x,z)J(z)\, dz\,, $$ under certain assumptions on $b$, $\kappa$ and $J$. The result allows more general coefficients even for $J(z)=|z|^{-d-1}$. - 11 April 2024
- Published by Wiley in Mathematische Nachrichten
- Vol. 297 (6) , 2234-2286
Nonsymmetric Lévy‐type operators
- 11 April 2024
- Published by Wiley in Mathematische Nachrichten
- Vol. 297 (6) , 2234-2286
Nonsymmetric Lévy‐type operators
- Open Access
- 14 February 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Documenta Mathematica
- Vol. 29 (1) , 209-245
Relativistic stable operators with critical potentials
We give local in time sharp two sided estimates of the heat kernel associated with the relativistic stable operator perturbed by a critical (Hardy) potential.
- 14 February 2024
- Published by European Mathematical Society - EMS - Publishing House GmbH in Documenta Mathematica
- Vol. 29 (1) , 209-245
Open AccessRelativistic stable operators with critical potentials
We give local in time sharp two sided estimates of the heat kernel associated with the relativistic stable operator perturbed by a critical (Hardy) potential.
- Open Access
Regularity of fundamental solutions for Lévy-type operators
For a class of non-symmetric non-local L\'evy-type operators $\mathcal{L}^{\kappa}$, which include those of the form $$ \mathcal{L}^{\kappa}f(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left)\kappa(x,z)J(z)\, dz\,, $$ we prove regularity of the fundamental solution $p^{\kappa}$ to the equation $\partial_t =\mathcal{L}^{\kappa}$.Open AccessRegularity of fundamental solutions for Lévy-type operators
For a class of non-symmetric non-local L\'evy-type operators $\mathcal{L}^{\kappa}$, which include those of the form $$ \mathcal{L}^{\kappa}f(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left)\kappa(x,z)J(z)\, dz\,, $$ we prove regularity of the fundamental solution $p^{\kappa}$ to the equation $\partial_t =\mathcal{L}^{\kappa}$. - Open Access
- 5 October 2023
- Published by Springer Nature in Journal d'Analyse Mathématique
- Vol. 152 (1) , 255-282
Sharp and plain estimates for Schrödinger perturbation of Gaussian kernel
We investigate whether a fundamental solution of the Schrödinger equation ∂tu = (Δ + V)u has local in time sharp Gaussian estimates. We compare that class with the class of V for which local in time plain Gaussian estimates hold. We concentrate on V that have fixed sign and we present certain conclusions for V in the Kato class.
- 5 October 2023
- Published by Springer Nature in Journal d'Analyse Mathématique
- Vol. 152 (1) , 255-282
Open AccessSharp and plain estimates for Schrödinger perturbation of Gaussian kernel
We investigate whether a fundamental solution of the Schrödinger equation ∂tu = (Δ + V)u has local in time sharp Gaussian estimates. We compare that class with the class of V for which local in time plain Gaussian estimates hold. We concentrate on V that have fixed sign and we present certain conclusions for V in the Kato class.
- Open Access
- 31 July 2023
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 403 (2) , 795-828
Bound States and Heat Kernels for Fractional-Type Schrödinger Operators with Singular Potentials
We consider non-local Schrödinger operators $$H=-L-V$$ in $$L^2(\mathbb {R}^d)$$ , $$d \geqslant 1$$ , where the kinetic terms L are pseudo-differential operators which are perturbations of the fractional Laplacian by bounded non-local operators and V is the fractional Hardy potential. We prove pointwise estimates of eigenfunctions corresponding to negative eigenvalues and upper finite-time horizon estimates for heat kernels. We also analyze the relation between the matching lower estimates of the heat kernel and the ground state near the origin. Our results cover the relativistic Schrödinger operator with Coulomb potential.- 31 July 2023
- Published by Springer Nature in Communications in Mathematical Physics
- Vol. 403 (2) , 795-828
Open AccessBound States and Heat Kernels for Fractional-Type Schrödinger Operators with Singular Potentials
We consider non-local Schrödinger operators $$H=-L-V$$ in $$L^2(\mathbb {R}^d)$$ , $$d \geqslant 1$$ , where the kinetic terms L are pseudo-differential operators which are perturbations of the fractional Laplacian by bounded non-local operators and V is the fractional Hardy potential. We prove pointwise estimates of eigenfunctions corresponding to negative eigenvalues and upper finite-time horizon estimates for heat kernels. We also analyze the relation between the matching lower estimates of the heat kernel and the ground state near the origin. Our results cover the relativistic Schrödinger operator with Coulomb potential. - Open Access
Relativistic stable operators with critical potentials
We give local in time sharp two sided estimates of the heat kernel associated with the relativistic stable operator perturbed by a critical (Hardy) potential.
Open AccessRelativistic stable operators with critical potentials
We give local in time sharp two sided estimates of the heat kernel associated with the relativistic stable operator perturbed by a critical (Hardy) potential.
- 1 January 2023
- Published by Institute of Mathematics, Polish Academy of Sciences in Bulletin of the Polish Academy of Sciences. Mathematics
- Vol. 71 (2) , 169-191
Regularity of fundamental solutions for Lévy-type operators
For a class of non-symmetric non-local Lévy-type operators $\mathcal L^{\kappa}$, which include those of the form $$ \mathcal L^{\kappa}f(x):= \int _{\mathbb R^d}( f(x+z)-f(x)- \mathbf 1_{|z| \lt 1} \langle z,\nabla f(x)\rangle )\kappa (x,z)J(z)\, dz, $$- 1 January 2023
- Published by Institute of Mathematics, Polish Academy of Sciences in Bulletin of the Polish Academy of Sciences. Mathematics
- Vol. 71 (2) , 169-191
Regularity of fundamental solutions for Lévy-type operators
For a class of non-symmetric non-local Lévy-type operators $\mathcal L^{\kappa}$, which include those of the form $$ \mathcal L^{\kappa}f(x):= \int _{\mathbb R^d}( f(x+z)-f(x)- \mathbf 1_{|z| \lt 1} \langle z,\nabla f(x)\rangle )\kappa (x,z)J(z)\, dz, $$