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Karol Szczypkowski

Publications and source records attributed to Karol Szczypkowski.

At least 19 recordsLinked to original sources
arXiv2026

Unifying Deep Stochastic Processes for Image Enhancement

Deep stochastic processes have recently become a central paradigm for image enhancement, with many methods explicitly conditioning the stochastic trajectory on the degraded input. However, the relationship between these conditional processes and standard diffusion models remains unclear. In this work, we introduce a unified perspective on stochastic image enhancement by classifying recent methods into three families of continuous-time processes: unconditional diffusion models, Ornstein-Uhlenbeck (OU) processes, and diffusion bridges. We show that all of these approaches arise from a common stochastic differential equation (SDE) formulation. This framework makes explicit that seemingly disparate methods differ primarily in their drift and diffusion terms, terminal distributions, and boundary conditions, while schedulers and samplers constitute orthogonal design choices. Leveraging this unification, we conduct a controlled empirical study across multiple image enhancement tasks using identical architectures and training protocols. Our results reveal no consistently dominant method; instead, we identify and disentangle the specific design choices that most strongly influence performance. Finally, we release ItoVision, a modular PyTorch library that implements the unified framework and enables rapid prototyping and fair comparison of stochastic image enhancement methods.

cs.CV↗
arXiv2025

Partial versus total resetting for Lévy flights in d dimensions: similarities and discrepancies

While stochastic resetting (or total resetting) is 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.

cond-mat.stat-mech↗
arXiv2024

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 $ρ_{\mathbf{Y}}(y)$.

math.PR↗
arXiv2024

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 $$ \mathcal{L}f(x):= b(x)\cdot \nabla f(x)+ \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left )κ(x,z)J(z)\, dz\,, $$ under certain assumptions on $b$, $κ$ and $J$. The result allows more general coefficients even for $J(z)=|z|^{-d-1}$.

math.AP↗
arXiv2023

Regularity of fundamental solutions for Lévy-type operators

For a class of non-symmetric non-local Lévy-type operators $\mathcal{L}^κ$, which include those of the form $$ \mathcal{L}^κf(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left )κ(x,z)J(z)\, dz\,, $$ we prove regularity of the fundamental solution $p^κ$ to the equation $\partial_t =\mathcal{L}^κ$.

math.AP↗
arXiv2022

Bound states and heat kernels for fractional-type Schrödinger operators with singular potentials

We consider non-local Schrödinger operators $H=-L-V$ in $L^2(\mathbf{R}^d)$, $d \geq 1$, 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.

math.FA↗
arXiv2021

Fundamental solution for super-critical non-symmetric Lévy-type operators

We prove the existence and give estimates of the fundamental solution (the heat kernel) for the equation $\partial_t =\mathcal{L}^κ$ for non-symmetric non-local operators $$ \mathcal{L}^κf(x):= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left )κ(x,z)J(z)\, dz\,, $$ under broad assumptions on $κ$ and $J$. Of special interest is the case when the order of the operator $\mathcal{L}^κ$ is smaller than or equal to 1. Our approach rests on imposing suitable cancellation conditions on the internal drift coefficient $$ \int_{r\leq |z|<1} z κ(x,z)J(z)dz\,,\qquad 0<r\leq 1\,, $$ which allows us to handle the non-symmetry of $z\mapsto κ(x,z)J(z)$. The results are new even for the $1$-stable Lévy measure $J(z)=|z|^{-d-1}$.

math.AP↗
arXiv2020

Sharp and plain estimates for Schrödinger perturbation of Gaussian kernel

We investigate whether a fundamental solution of the Schrödinger equation $\partial_t u =(Δ+V)\, u$ 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.

math.AP↗
arXiv2020

Estimates of heat kernels of non-symmetric Lévy processes

We investigate densities of vaguely continuous convolution semigroups of probability measures on $\mathbb{R}^d$. First, we provide results that give upper estimates in a situation when the corresponding jump measure is allowed to be highly non-symmetric. Further, we prove upper estimates of the density and its derivatives if the jump measure compares with an isotropic unimodal measure and the characteristic exponent satisfies certain scaling condition. Lower estimates are discussed in view of a recent development in that direction, and in such a way to complement upper estimates. We apply all those results to establish precise estimates of densities of non-symmetric Lévy processes.

math.PR↗
arXiv2019

L{é}vy processes: concentration function and heat kernel bounds

We investigate densities of vaguely continuous convolution semigroups of probability measures on $\mathbb{R}^d$. We expose that many typical conditions on the characteristic exponent repeatedly used in the literature of the subject are equivalent to the behaviour of the maximum of the density as a function of time variable. We also prove qualitative lower estimates under mild assumptions on the corresponding jump measure and the characteristic exponent.

math.PR↗
arXiv2018

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.

math.FA↗
arXiv2018

Heat kernels of non-symmetric Lévy-type operators

We construct the fundamental solution (the heat kernel) $p^κ$ to the equation $\partial_t=\mathcal{L}^κ$, where under certain assumptions the operator $\mathcal{L}^κ$ takes one of the following forms, \begin{align*} \mathcal{L}^κf(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x)- 1_{|z|<1} \left )κ(x,z)J(z)\, dz \,, \mathcal{L}^κf(x)&:= \int_{\mathbb{R}^d}( f(x+z)-f(x))κ(x,z)J(z)\, dz\,, \mathcal{L}^κf(x)&:= \frac1{2}\int_{\mathbb{R}^d}( f(x+z)+f(x-z)-2f(x))κ(x,z)J(z)\, dz\,. \end{align*} In particular, $J\colon \mathbb{R}^d \to [0,\infty]$ is a Lévy density, i.e., $\int_{\mathbb{R}^d}(1\land |x|^2)J(x)dx<\infty$. The function $κ(x,z)$ is assumed to be Borel measurable on $\mathbb{R}^d\times \mathbb{R}^d$ satisfying $0<κ_0\leq κ(x,z)\leq κ_1$, and $|κ(x,z)-κ(y,z)|\leq κ_2|x-y|^β$ for some $β\in (0, 1)$. We prove the uniqueness, estimates, regularity and other qualitative properties of $p^κ$.

math.AP↗
arXiv2016

Kato classes for Lévy processes

We prove that the definitions of the Kato class by the semigroup and by the resolvent of the Lévy process on $\mathbb{R}^d$ coincide if and only if 0 is not regular for {0}. If 0 is regular for {0} then we describe both classes in detail. We also give an analytic reformulation of these results by means of the characteristic (Lévy-Khintchine) exponent of the process. The result applies to the time-dependent (non-autonomous) Kato class. As one of the consequences we obtain a simultaneous time-space smallness condition equivalent to the Kato class condition given by the semigroup.

math.FA↗
arXiv2015

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.

math.FA↗
arXiv2014

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.

math.AP↗