Tomasz Klimsiak

activity

20092026

most citedDirichlet forms and semilinear elliptic equations with measure data

53 citations · 217 across the 49 of their papers we have counts for

collaborators

59 papers

math.AP2026

Stability of Solutions to Semilinear Exterior Dirichlet Problems under Convergence of Lévy Symbols

Tomasz Klimsiak, Andrzej Rozkosz, Leszek Słomiński

We study stability of semilinear exterior Dirichlet problems driven by possibly nonsymmetric Levy operators. Let $L^{n}$ and $L$ have Lévy-Khintchine symbols $ψ_{n}$ and $ψ$ , respectiv…

math.PR2026

A Quantitative Characterization of the Mokobodzki Condition

Tomasz Klimsiak, Maurycy Rzymowski

We establish a quantitative characterization of the Mokobodzki condition for two adapted càdlàg barriers on an arbitrary filtered probability space. For every $p \geq 1$ , we introduc…

math.AP2026

Isolated Singularities and Measure Data Problems for Semilinear Equations Driven by Stable Lévy Operators

Kamil Dunst, Tomasz Klimsiak

We investigate positive solutions of semilinear equations driven by uniformly elliptic strictly $2 s$ -stable Lévy operators, where $s \in (0, 1)$ . We first prove that every positive d…

math.AP2026

Positive Solutions for Sublinear Equations with Compact Positivity-Improving Resolvent

Tomasz Klimsiak

We establish a Brézis-Oswald-type spectral principle for semilinear equations \[ -Lu=f(x,u), \] where $L$ is the generator of a positive $C_{0}$ -semigroup on $L^{p}$ with compact posit…

math.AP2025

Nonlinear Hardy-Stein type identities for harmonic functions relative to symmetric integro-differential operators

Tomasz Klimsiak, Andrzej Rozkosz

We show identities of Hardy-Stein type for harmonic functions relative to integro-differential operators corresponding to general symmetric regular Dirichlet forms satisfying the a…

math.AP2025

Bôcher type theorem for elliptic equations with drift perturbed Lévy operator

Tomasz Klimsiak

A classical Bôcher's theorem asserts that any positive harmonic function (with respect to the Laplacian) in the punctured unit ball can be expressed, up to the multiplication const…

math.PR2024

Mokobodzki's intervals: an approach to Dynkin games when value process is not a semimartingale

Tomasz Klimsiak, Maurycy Rzymowski

We study Dynkin games governed by a nonlinear $E^{f}$ -expectation on a finite interval $[0, T]$ , with payoff càdlàg processes $L, U$ of class (D) which are not imposed to satis…

math.AP2024

Poisson equation with measure data, reconstruction formula and Doob classes of processes

Andrzej Rozkosz, Tomasz Klimsiak

We consider the Dirichlet problem for equation involving a general operator associated with a symmetric transient regular Dirichlet form and bounded Borel measure on the right-hand…

math.PR2024

Homogenization of stable-like operators with random, ergodic coefficients

Tomasz Klimsiak, Tomasz Komorowski, Lorenzo Marino

We show homogenization for a family of $R^{d}$ -valued stable-like processes $(X_{t}^{ε; θ})_{t \geq 0}$ , $ε \in (0, 1]$ , whose (random) Fourier symbols equal $q_ε(x,ξ;θ)=\frac{1}{ε^…

math.AP2023

Generalized solutions to semilinear elliptic equations with measure data

Tomasz Klimsiak

We address an open problem posed by H. Brezis, M. Marcus and A.C. Ponce in: Nonlinear elliptic equations with measures revisited. In: Mathematical Aspects of Nonlinear Dispersive E…

math.AP2023★ 6 cited

Dirichlet problem for semilinear partial integro-differential equations: the method of orthogonal projection

Tomasz Klimsiak, Andrzej Rozkosz

We study the Dirichlet problem for semilinear equations on general open sets with measure data on the right-hand side and irregular boundary data. For this purpose we develop the c…

math.PR2022

A priori estimates for multidimensional BSDEs with integrable data

Tomasz Klimsiak, Maurycy Rzymowski

We study Backward Stochastic Differential Equations on a probability space equipped with a Brownian filtration. We assume that the terminal value and the generator at zero are mere…

math.PR2022

Nonlinear BSDEs with two optional Doob's class barriers satisfying weak Mokobodzki's condition and extended Dynkin games

Tomasz Klimsiak, Maurycy Rzymowski

We study reflected backward stochastic differential equation (RBSDEs) on the probability space equipped with a Brownian motion. The main novelty of the paper lies in fact that we c…

math.AP2021

Nonlinear elliptic equations with integro-differential divergence form operators and measure data under sign condition on the nonlinearity

Tomasz Klimsiak

We study existence problem for semilinear equations with Borel measure data and operator generated by a symmetric Markov semigroup. We assume merely that the nonlinear part satisfi…

math.PR2021

Nonlinear BSDEs in general filtration with drivers depending on the martingale part of a solution

Tomasz Klimsiak, Maurycy Rzymowski

In the present paper, we consider multidimensional nonlinear backward stochastic differential equations (BSDEs) with a driver depending on the martingale part $M$ of a solution. We…

math.PR2021

Hopf type lemmas for subsolutions of integro-differential equations

Tomasz Klimsiak, Tomasz Komorowski

In the paper we prove a generalization of the Hopf lemma for weak subsolutions of the equation: $- A u + c u = 0$ in $D$ , for a wide class of Lévy type integro-differential operators $A$ ,…

math.AP2020

On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators

Tomasz Klimsiak, Tomasz Komorowski

In the present paper we prove estimates on {subsolutions of the equation $- A v + c (x) v = 0$ }, $x \in D$ , where $D \subset R^{d}$ is a domain (i.e. an open and connected set) and $A$ is a…

math.PR2020

Long-time asymptotic behaviour of the value function in nonlinear stopping problems

Tomasz Klimsiak, Andrzej Rozkosz

We provide general conditions ensuring that the value functions of some nonlinear stopping problems with finite horizon converge to the value functions of the corresponding problem…

math.AP2020

On uniqueness and structure of renormalized solutions to integro-differential equations with general measure data

Tomasz Klimsiak

We propose a new definition of renormalized solution to linear equation with self-adjoint operator generating a Markov semigroup and bounded Borel measure on the right-hand side. W…

math.PR2020

Reflected BSDEs with two optional barriers and monotone coefficient on general filtered space

Tomasz Klimsiak, Maurycy Rzymowski

We consider reflected backward stochastic differential equations with two optional barriers of class (D) satisfying Mokobodzki's separation condition and coefficient which is only…

math.AP2019

Location of zeros of non-trivial positive supersolutions to Schrödinger equations

Tomasz Klimsiak

We study Schrödinger operators on $L^{2} (E; m)$ of the form $- A + V$ with singular potentials $V$ . We address the question posed by H. Brezis about the structure of the set ${u = 0}$ fo…

math.AP2019

Schrödinger equations with smooth measure potential and general measure data

Tomasz Klimsiak

We study equations driven by Schrödinger operators consisting of a self-adjoint Dirichlet operator and a singular potential, which belongs to a class of positive Borel measures abs…

math.AP2019

Asymptotics for logistic-type equations with Dirichlet fractional Laplace operator

Tomasz Klimsiak

We study the asymptotics of solutions of logistic type equations with fractional Laplacian as time goes to infinity and as the exponent in nonlinear part goes to infinity. We prove…

math.AP2019

Uniqueness for an obstacle problem arising from logistic-type equations with fractional Laplacian

Tomasz Klimsiak

We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under considerati…