arXiv subjects
Publications and source records attributed to Łukasz Leżaj.
We construct a self-similar solution of the heat equation for the fractional Laplacian with Dirichlet boundary conditions in every fat cone. As applications, we give the Yaglom limit and entrance law for the corresponding killed isotropic stable Lévy process and precise large-time asymptotics for solutions of the Cauchy problem in the cone.
We study a $d$-dimensional non-symmetric strictly $α$-stable Lévy process $\mathbf{X}$, whose spherical density is bounded and bounded away from the origin. First, we give sharp two-sided estimates on the transition density of $\mathbf{X}$ killed when leaving an arbitrary $κ$-fat set. We apply these results to get the existence of the Yaglom limit for arbitrary $κ$-fat cone. In the meantime we also obtain the spacial asymptotics of the survival probability at the vertex of the cone expressed by means of the Martin kernel for $Γ$ and its homogeneity exponent. Our results hold for the dual process $\widehat{\mathbf{X}}$, too.
We prove universality of the Yaglom limit of Lipschitz cones among all unimodal Lévy processes sufficiently close to the isotropic $α$-stable Lévy process.
We prove existence and asymptotic behavior of the transition density for a large class of subordinators whose Laplace exponents satisfy lower scaling condition at infinity. Furthermore, we present lower and upper bounds for the density. Sharp estimates are provided if additional upper scaling condition on the Laplace exponent is imposed. In particular, we cover the case when the (minus) second derivative of the Laplace exponent is a function regularly varying at infinity with regularity index bigger than -2.
We prove asymptotic behaviour of transition density for a large class of spectrally one-sided Lévy processes of unbounded variation satisfying mild condition imposed on the second derivative of the Laplace exponent, or equivalently, on the real part of the characteristic exponent. We also provide sharp two-sided estimates on the density when restricted additionally to processes without Gaussian component.
We prove sharp two-sided estimates on the tail probability of the first hitting time of bounded interval as well as its asymptotic behaviour for general non-symmetric processes which satisfy an integral condition \[ \int_0^{\infty} \frac{dξ}{1+\operatorname{Re} ψ(ξ)}<\infty. \] To this end, we first prove and then apply the global scale invariant Harnack inequality. Results are obtained under certain conditions on the characteristic exponent. We provide a wide class of Lévy processs which satisfy these assumptions.
We study translation-invariant integrodifferential operators that generate Lévy processes. First, we investigate different notions of what a solution to a nonlocal Dirichlet problem is and we provide the classical representation formula for distributional solutions. Second, we study the question under which assumptions distributional solutions are twice differentiable in the classical sense. Sufficient conditions and counterexamples are provided.