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Tomasz Klimsiak

Publications and source records attributed to Tomasz Klimsiak.

At least 37 records · Page 2Linked to original sources
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↗
arXiv2021

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

We prove a uniqueness theorem for the obstacle problem for linear equations involving the fractional Laplacian with zero Dirichlet exterior condition. The problem under consideration arises as the limit of some logistic-type equations. Our result extends (and slightly strengthens) the known corresponding results for the classical Laplace operator with zero boundary condition. Our proof, as compared with the known proof for the classical Laplace operator, is entirely new, and is based on the probabilistic potential theory. Its advantage is that it may be applied to a wide class of integro-differential operators.

math.AP↗
arXiv2021

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

In the present paper we prove estimates on {subsolutions of the equation $-Av+c(x)v=0$}, $x\in D$, where $D\subset \bbR^d$ is a domain (i.e. an open and connected set) and $A$ is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable. Some of our results may be considered "quantitative" versions of the Hopf lemma, as they provide the lower bound on the outward normal directional derivative at the maximum point of a subsolution %on a boundary of a domain in terms of its value at the point. We shall also show lower bounds on the subsolution around its maximum point by the principal eigenfunction associated with $A$ and the domain. Additional results, among them the weak and strong maximum principles, the weak Harnack inequality are also proven.

math.AP↗
arXiv2021

Quasi-regular Dirichlet forms and the obstacle problem for elliptic equations with measure data

We consider the obstacle problem with irregular barriers for semilinear elliptic equation involving measure data and operator corresponding to a general quasi-regular Dirichlet form. We prove existence and uniqueness of a solution as well as its representation as an envelope of a supersolution to some related partial differential equation. We also prove regularity results for the solution and the Lewy-Stampacchia inequality.

math.PR↗
arXiv2021

Non-semimartingale solutions of reflected BSDEs and applications to Dynkin games

We introduce a new class of reflected backward stochastic differential equations with two càdlàg barriers, which need not satisfy any separation conditions. For that reason, in general, the solutions are not semimartingales. We prove existence, uniqueness and approximation results for solutions of equations defined on general filtered probability spaces. Applications to Dynkin games and variational inequalities, both stationary and evolutionary, are given.

math.PR↗
arXiv2021

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

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 assume that the nonlinear term is merely monotone continuous with respect to the state variable. As to the regularity of the driver with respect to the martingale variable, we consider a very general condition which permits path-dependence on "the future" of the process $M$ as well as a dependence of its law (McKean-Vlasov-type equations). For such driver, we prove the existence and uniqueness of a global solution (i.e. for any maturity $T>0$) to BSDE with data satisfying natural integrability conditions.

math.PR↗
arXiv2020

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

We consider reflected backward stochastic differential equations with two optional barriers of class (D) satisfying Mokobodzki's separation condition and coefficient which is only continuous and non-increasing. We assume that data are merely integrable and the terminal time is an arbitrary (possibly infinite) stopping time. We study the problem of existence and uniqueness of solutions, and their connections with the value process in nonlinear Dynkin games.

math.PR↗
arXiv2019

On the structure of diffuse measures for parabolic capacities

Let $Q=(0,T)\timesΩ$, where $Ω$ is a bounded open subset of $\mathbb{R}^d$. We consider the parabolic $p$-capacity on $Q$ naturally associated with the usual $p$-Laplacian. Droniou, Porretta and Prignet have shown that if a bounded Radon measure $μ$ on $Q$ is diffuse, i.e. charges no set of zero $p$-capacity, $p>1$, then it is of the form $μ=f+\mbox{div}(G)+g_t$ for some $f\in L^1(Q)$, $G\in (L^{p'}(Q))^d$ and $g\in L^p(0,T;W^{1,p}_0(Ω)\cap L^2(Ω))$. We show the converse of this result: if $p>1$, then each bounded Radon measure $μ$ on $Q$ admitting such a decomposition is diffuse.

math.AP↗
arXiv2019

Renormalized solutions of semilinear elliptic equations with general measure data

In the paper, we first propose a definition of renormalized solution of semilinear elliptic equation involving operator corresponding to a general (possibly nonlocal) symmetric regular Dirichlet form satisfying the so-called absolute continuity condition and general (possibly nonsmooth) measure data. Then we analyze the relationship between our definition and other concepts of solutions considered in the literature (probabilistic solutions, solution defined via the resolvent kernel of the underlying Dirichlet form, Stampacchia's definition by duality). We show that under mild integrability assumption on the data all these concepts coincide.

math.AP↗
arXiv2018

Reflected backward stochastic differential equations with two optional barriers

We consider reflected backward stochastic differential equations with two general optional barriers. The solutions to these equations have the so-called regulated trajectories, i.e trajectories with left and right finite limits. We prove the existence and uniqueness of $\mathbb L^p$ solutions, $p\geq 1$, and show that the solutions may be approximated by a modified penalization method.

math.PR↗
arXiv2018

Reflected BSDEs and obstacle problem for semilinear PDEs in divergence form

We consider the Cauchy problem for semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle the problem has a unique solution and we provide its stochastic representation in terms of reflected backward stochastic differential equations. We prove also regularity properties and approximation results for solutions of the problem.

math.AP↗
arXiv2018

BSDEs with monotone generator and two irregular reflecting barriers

We consider BSDEs with two reflecting irregular barriers. We give necessary and sufficient conditions for existence and uniqueness of $\mathbb{L}^{p}$ solutions for equations with generators monotone with respect to $y$ and Lipschitz continuous with respect to $z$, and with data in $\mathbb{L}^{p}$ spaces for $p\ge 1$. We also prove that the solutions can be approximated via penalization method.

math.PR↗
arXiv2018

Reflected BSDEs with monotone generator

We give necessary and sufficient condition for existence and uniqueness of $\mathbb{L}^{p}$-solutions of reflected BSDEs with continuous barrier, generator monotone with respect to $y$ and Lipschitz continuous with respect to $z$, and with data in $\mathbb{L}^{p}$, $p\ge 1$. We also prove that the solutions may be approximated by the penalization method.

math.PR↗
arXiv2018

Obstacle problem for evolution equations involving measure data and operator corresponding to semi-Dirichlet form

In the paper, we consider the obstacle problem, with one and two irregular barriers, for semilinear evolution equation involving measure data and operator corresponding to a semi-Dirichlet form. We prove the existence and uniqueness of solutions under the assumption that the right-hand side of the equation is monotone and satisfies mild integrability conditions. To treat the case of irregular barriers, we extend the theory of precise versions of functions introduced by M. Pierre. We also give some applications to the so-called switching problem.

math.AP↗
arXiv2018

Smooth measures and capacities associated with nonlocal parabolic operators

We consider a family $\{L_t,\, t\in [0,T]\}$ of closed operators generated by a family of regular (non-symmetric) Dirichlet forms $\{(B^{(t)},V),t\in[0,T]\}$ on $L^2(E;m)$. We show that a bounded (signed) measure $μ$ on $(0,T)\times E$ is smooth, i.e. charges no set of zero parabolic capacity associated with $\frac{\partial}{\partial t}+L_t$, if and only if $μ$ is of the form $μ=f\cdot m_1+g_1+\partial_tg_2$ with $f\in L^1((0,T)\times E;dt\otimes m)$, $g_1\in L^2(0,T;V')$, $g_2\in L^2(0,T;V)$. We apply this decomposition to the study of the structure of additive functionals in the Revuz correspondence with smooth measures. As a by-product, we also give some existence and uniqueness results for solutions of semilinear equations involving the operator $\frac{\partial}{\partial t}+L_t$ and a functional from the dual $\mathcal{W}'$ of the space $\mathcal{W}=\{u\in L^2(0,T;V):\partial_t u\in L^2(0,T;V')\}$ on the right-hand side of the equation.

math.AP↗
arXiv2018

On semilinear elliptic equations with diffuse measures

We consider semilinear equation of the form $-Lu=f(x,u)+μ$, where $L$ is the operator corresponding to a transient symmetric regular Dirichlet form ${\mathcal E}$, $μ$ is a diffuse measure with respect to the capacity associated with ${\mathcal E}$, and the lower-order perturbing term $f(x,u)$ satisfies the sign condition in $u$ and some weak integrability condition (no growth condition on $f(x,u)$ as a function of $u$ is imposed). We prove the existence of a solution under mild additional assumptions on ${\mathcal E}$. We also show that the solution is unique if $f$ is nonincreasing in $u$.

math.AP↗
arXiv2018

Large time behaviour of solutions to parabolic equations with Dirichlet operators and nonlinear dependence on measure data

We study large time behaviour of solutions of the Cauchy problem for equations of the form $\partial_tu-L u+λu=f(x,u)+g(x,u)\cdotμ$, where $L$ is the operator associated with a regular lower bounded semi-Dirichlet form ${\mathcal{E}}$ and $μ$ is a nonnegative bounded smooth measure with respect to the capacity determined by ${\mathcal{E}}$. We show that under the monotonicity and some integrability assumptions on $f,g$ as well as some assumptions on the form ${\mathcal{E}}$, $u(t,x)\rightarrow v(x)$ as $t\rightarrow\infty$ for quasi-every $x$, where $v$ is a solution of some elliptic equation associated with our parabolic equation. We also provide the rate convergence. Some examples illustrating the utility of our general results are given.

math.AP↗