Searcharxiv⌕ Search

arXiv subjects

Tomasz Klimsiak

Publications and source records attributed to Tomasz Klimsiak.

At least 19 recordsLinked to original sources
arXiv2026

Positive Solutions for Sublinear Equations with Compact Positivity-Improving Resolvent

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 positivity-improving resolvent. Neither symmetry, variational structure, nor regularizing properties such as ultracontractivity or smoothing are assumed. Let $a_0$ and $a_\infty$ denote the asymptotic slopes of the nonlinearity at zero and at infinity, and let $λ_1(a)$ denote the generalized principal eigenvalue associated with the perturbed operator $-L-a$. We prove that \[ λ_1(a_0)<0<λ_1(a_\infty) \] implies that the equation admits a strictly positive solution. The proof develops a potential-theoretic sub- and supersolution framework based on the order induced by supermedian functions and combines a Deny-type compactness theorem, a Kato-type inequality, and a Doob transform. The construction also yields an order-preserving selection of solutions and allows the lower control on the nonlinearity to be relaxed. If $y\mapsto f(x,y)/y$ is strictly decreasing and $a_0$ is bounded, the spectral condition is necessary as well, and the strictly positive solution is unique.

math.AP↗
arXiv2026

A Quantitative Characterization of the Mokobodzki Condition

We establish a quantitative characterization of the Mokobodzki condition for two adapted càdlàg barriers on an arbitrary filtered probability space. For every $p\geq1$, we introduce a mean co-variation functional $Var_p(L,U)$, which for $p=1$ and $L=U$ reduces to Rao's mean variation, and show that it is quantitatively equivalent to the minimal $\underline H^p$-norm among all semimartingales lying between the barriers. The result covers both the case $p>1$ and the endpoint $p=1$, where the natural scale is based on Doob's class $(D)$.

math.PR↗
arXiv2026

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

We investigate positive solutions of semilinear equations driven by uniformly elliptic strictly $2s$-stable Lévy operators, where $s\in (0,1)$. We first prove that every positive distributional solution of $-Lu=u^p$ in a punctured domain $D\setminus\{0\}$ satisfies $-Lu=u^p+kδ_0$ in $D$ for some $k\ge0$, and that necessarily $k=0$ whenever $p\ge d/(d-2s)$. We then study the corresponding Dirichlet problem in which the Dirac mass is replaced by a bounded positive measure, and establish the existence of a critical parameter $k_μ$: below this threshold minimal positive solutions exist, whereas above it the problem admits no solution. In the symmetric case, we further prove multiplicity below the threshold, as well as existence and uniqueness at the threshold itself.

math.AP↗
arXiv2025

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

We show identities of Hardy-Stein type for harmonic functions relative to integro-differential operators corresponding to general symmetric regular Dirichlet forms satisfying the absolute continuity condition. The novelty is that we consider operators of mixed type containing both local and nonlocal component. Moreover, the identities are proved for compositions of harmonic functions and general convex functions. We also provide some conditional identities, i.e. identities for ratios of harmonic functions. As an application we give a characterization of norms in harmonic Hardy spaces and prove Littlewood--Paley type estimates for square functions. To illustrate general results, we discuss in some details the case of divergence form operator and purely nonlocal operator defined by some jump kernel. Our proofs are rather short and use mainly probabilistic methods.

math.AP↗
arXiv2025

Bôcher type theorem for elliptic equations with drift perturbed Lévy operator

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 constant, as the sum of the Newtonian kernel and a positive function that is harmonic in the whole unit ball. This theorem expresses one of the fundamental results in the theory of isolated singularities and it can be viewed as a statement on the asymptotic behavior of positive harmonic functions near their isolated singularities. In the paper we generalize this results to drift perturbed Lévy operators. We propose a new approach based on the probabilistic potential theory. It applies to Lévy operators for which the resolvent of its perturbation is strongly Feller. In particular our result encompasses drift perturbed fractional Laplacians with any stability index bounded between zero and two - the method therefore applies to subcritical and supercritical cases.

math.AP↗
arXiv2024

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

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 classical method of orthogonal projection. We treat in a unified form equations with operators belonging to the broad class of integro-differential operators associated with symmetric regular Dirichlet forms.

math.AP↗
arXiv2024

Homogenization of stable-like operators with random, ergodic coefficients

We show homogenization for a family of $\mathbb{R}^d$-valued stable-like processes $(X_t^{ε;θ})_{t\ge 0}$, $ε\in(0,1]$, whose (random) Fourier symbols equal $q_ε(x,ξ;θ)=\frac{1}{ε^α}q(x/ε,εξ; θ)$, where$$q(x,ξ; θ)=\int_{\mathbb{R}^d}\big(1-e^{i y\cdotξ}+iy\cdotξ\mathds{1}_{\{|y|\le1\}}\big)\,\frac{\langle a(x;θ)y,y\rangle}{|y|^{d+2+α}}\,dy,$$for $(x,ξ,θ)\in\mathbb{R}^{2d}\timesΘ$. Here, $α\in(0,2)$ and the family $(a(x; θ))_{x\in\mathbb{R}^d}$ of $d\times d$ symmetric, non-negative definite matrices is a stationary ergodic random field over some probability space $(Θ,{\cal H},m)$. We assume that the random field is deterministically bounded and non-degenerate, i.e.\ $|a(x;θ)|\leΛ$ and $\text{Tr}(a(x;θ))\geλ$ for some $Λ,λ>0$ and all $θ\inΘ$. In addition, we suppose that the field is regular enough so that for any $θ\inΘ$, the operator $-q(\cdot,D;θ)$, defined on the space of compactly supported $C^2$ functions, is closable in the space of continuous functions vanishing at infinity and its closure generates a Feller semigroup. We prove the weak convergence of the laws of $(X_t^{ε;θ})_{t\ge 0}$, as $ε\to0^+$, in the Skorokhod space, $m$-a.s.\ in $θ$, to an $α$-stable process whose Fourier symbol $\bar{q}(ξ)$ is given by $\bar{q}(ξ)=\int_Ωq(0,ξ;θ)Φ_*(θ)\,m(dθ)$, where $Φ_*$ is a strictly positive density w.r.t.\ measure $m$. Our result has an analytic interpretation in terms of the convergence, as $ε\to0^+$, of the solutions to random integro-differential equations $ \partial_tu_ε(t,x;θ)=-q_ε(x,D;θ)u_ε(t,x;θ)$, with the initial condition $u_ε(0,x;θ)=f(x)$, where $f$ is a bounded and continuous function.

math.PR↗
arXiv2024

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

We study Dynkin games governed by a nonlinear $\mathbb E^f$-expectation on a finite interval $[0,T]$, with payoff càdlàg processes $L,U$ of class (D) which are not imposed to satisfy (weak) Mokobodzki's condition - the existence of a càdlàg semimartingale between the barriers. For that purpose we introduce the notion of Mokobodzki's stochastic intervals $\mathscr M(θ)$ (roughly speaking, maximal stochastic interval on which Mokobodzki's condition is satisfied when starting from the stopping time $θ$) and the notion of reflected BSDEs without Mokobodzki's condition (this is a generalization and modification of the notion introduced by Hamadéne and Hassani (2005)). We prove an existence and uniqueness result for RBSDEs with driver $f$ that is non-increasing with respect to the value variable (no restrictions on the growth) and Lipschitz continuous with respect to the control variable, and with data in $L^1$ spaces. Next, by using RBSDEs, we show numerous results on Dynkin games: existence of the value process, saddle points, and convergence of the penalty scheme. We also show that the game is not played beyond $\mathscr M(θ)$, when starting from $θ$.

math.PR↗
arXiv2024

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

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 side of the equation. We introduce a new function space (depending on the form) which allows us to distinguish between solutions with diffuse measure and with general Borel measure. This new space can by characterized analytically in terms of the Poisson kernel associated with the underlying operator or probabilistically by using the notion of Doob class (D) of processes naturally associated with the operator. We also prove a reconstruction formula describing, in terms of the carré du champ operator and jump measure associated with the underlying form, the behaviour of the solution on the set where it is very large.

math.AP↗
arXiv2023

Generalized solutions to semilinear elliptic equations with measure data

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 Equations (J. Bourgain, C. Kenig, S. Klainerman, eds.), Annals of Mathematics Studies, 163 (2007). We prove that for any bounded Borel measure $μ$ on a smooth bounded domain $D\subset\mathbb R^d$ and asymptotically convex non-decreasing non-negative continuous function $g$ on $\mathbb R$ the sequence of solutions to the semi-linear equation (P): $-Δu+g(u)=ρ_n\astμ$ ($ρ_n$ is a mollifier) that is subject to homogeneous Dirichlet condition, converges to the function that solves (P) with $ρ_n\astμ$ replaced by the reduced measure $μ^*$ (metric projection onto the space of good measures). We also provide a corresponding version of this result without non-negativity assumption on $g$.

math.AP↗
arXiv2022

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

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 $\{u=0\}$ for non-negative supersolutions to $-Au+Vu=0$. The class of operators $A$ we study in the paper includes, in particular, symmetric Levy type operators and symmetric diffusions in divergence form, with strictly positive Green functions. The class of potentials $V$ consists of positive smooth measures, which contains, in particular, Coulomb potentials and harmonic potentials, as well as generalized potentials, i.e. positive Borel measures concentrated on $m$-negligible sets.

math.AP↗
arXiv2022

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

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 problems with infinite horizon. Our result can be formulated as result on stability, with respect to time horizon, of nonlinear $f$-expectations. We also study the rate of convergence. Many examples are given to illustrate our results. They include the analysis of time asymptotics of the fair prices of American options in a multidimensional exponential Lévy model.

math.PR↗
arXiv2022

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

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 consider the following weak assumptions on the data: barriers are optional of class (D) satisfying weak Mokobodzki's condition, generator is continuous and non-increasing with respect to the value-variable (no restriction on the growth) and Lipschitz continuous with respect to the control-variable, and the terminal condition and the generator at zero are supposed to be merely integrable. We prove that under these conditions on the data there exists a solution to corresponding RBSDE. In the second part of the paper, we apply the theory of RBSDEs to solve basic problems in Dynkin games driven by nonlinear expectation based on the generator mentioned above. We prove that the main component of a solution to RBSDE represents the value process in corresponding extended nonlinear Dynkin game. Moreover, we provide sufficient condition on the barriers guaranteeing the existence of the value for nonlinear Dynkin games and the existence of a saddle point.

math.PR↗
arXiv2022

A priori estimates for multidimensional BSDEs with integrable data

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 merely integrable. Moreover, the generator is assumed to be non-increasing with respect to the value variable (with no restrictions on the growth) and Lipschitz continuous, with sublinear growth, with respect to the control variable. We provide a priori estimate and stability result for solutions to the aforementioned BSDEs.

math.PR↗
arXiv2022

Hopf type lemmas for subsolutions of integro-differential equations

In the paper we prove a generalization of the Hopf lemma for weak subsolutions of the equation: $-Au+cu=0$ in $D$, for a wide class of Lévy type integro-differential operators $A$, bounded and measurable function $c:D\to[0,+\infty)$ and domain $D\subset \BR^d$. More precisely, we prove that if {\em the strong maximum principle} (SMP) holds for $A$, then there exists a Borel function $ψ:D\to(0,+\infty)$, depending only on the coefficients of the operator $A$, $c$ and $D$ such that for any subsolution $u(\cdot)$ one can find a constant $a>0$ (that in general depends on $u$), for which $\sup_{y\in {\rm cl}D\cup {\cal S}(D)}u(y)-u(x)\ge aψ(x)$, $x\in D$. Here ${\rm cl}D$ is the closure of $D$. The set ${\cal S}(D)$ - called the range of non-locality of $A$ over $D$ - is determined by the support of the Levy jump measure associated with $A$. This type of a result we call the {\em generalized Hopf lemma}. It turns out that the irreducibility property of the resolvent of $A$ implies the SMP. The converse also holds, provided we assume some additional (rather weak) assumption on the resolvent. For some classes of operators we can admit the constant $a$ to be equal to $ \sup_{y\in {\rm cl}D\cup {\cal S}(D)}u(y)$. We call this type of a result a {\em quantitative version} of the Hopf lemma. Finally, we formulate a necessary and sufficient condition on $A$ - expressed in terms of ergodic properties of its resolvent - which ensures that the lower bound $ψ\in {\rm span}(φ_D)$, where $φ_D$ is a non-negative eigenfunction of $A$ in $D$. We show that the aforementioned ergodic property is implied by the intrinsic ultracontractivity of the semigroup associated with $A$.

math.PR↗
arXiv2021

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

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 satisfies the so-called sign condition. Using the method of sub and supersolutions we show the existence of maximal measure for which there exists a solution to the problem (the so-called reduced measure introduced by H. Brezis, M. Marcus and A.C. Ponce).

math.AP↗
arXiv2021

Schrödinger equations with smooth measure potential and general measure data

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 absolutely continuous with respect to a capacity generated by the operator. In particular, we cover positive potentials exploding on a set of capacity zero. The right-hand side of equations is allowed to be a general bounded Borel measure. The class of self-adjoint Dirichlet operators is quite large. Examples include integro-differential operators with the local part of divergence form. We give a necessary and sufficient condition for the existence of a solution, and prove some regularity and stability results.

math.AP↗
arXiv2021

Asymptotics for logistic-type equations with Dirichlet fractional Laplace operator

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 strong convergence of solutions in the energy space and uniform convergence to the solution of an obstacle problem. As a by-product, we also prove the cut-off property for eigenvalues of the Dirichlet fractional Laplace operator perturbed by exploding potentials.

math.AP↗