Martin representation and Relative Fatou Theorem for fractional Laplacian with a gradient perturbation

Let $L=\Delta^{\alpha/2}+ b\cdot\nabla$ with $\alpha\in(1,2)$. We prove the Martin representation and the Relative Fatou Theorem for non-negative singular $L$-harmonic functions on ${\mathcal C}^{1,1}$ bounded open sets.Comment: 28 pages, editorial change

What works for disadvantaged unemployed : private or public ALMP services? Evidence from Poland

Purpose: This paper aims to assess the effectiveness of public vs. private active labour market policies (ALMP) for disadvantaged unemployed. The literature on the efficiency of contracting-out ALMP services by public institutions is not consistent. Formalism and limited scope of possible actions in public institutions stand in the way to activation of the less promising unemployed. On the other hand, contracted-out companies work as a black-box and are paid a success fee. Design/Methodology/Approach: We used (conducted by accident) Randomised Control Trial evaluation of the contracting-out ALMP services program in Poland. We used public registers data from Poland. Findings: We found a strong positive effect of support provided by contracted private agencies. This effect was found to be the strongest among the most dismissed groups of the unemployed. Practical Implications: Study shows that success-fee for external institutions and lack of hard frames assures incentives compatibility that results in real improvement of the unemployed situation, even a year after the end of the intervention. This conclusion should result in adjustments in active labour market policy. Originality/Value: The uniqueness of the study lies in scale of the RCT experiment and robustness of conculusons. The golden standard of evaluation allowed for gaining indisputable evidence on the effectiveness of particular ALMP instrument. Without this kind of analysis, public decisions can turn out to be wrong and lead to the closure of effective programmes (as it happened with contracting-out ALMP services), and investment in programmes that do not provide any effective support for those in need.peer-reviewe

On Wiener-Hopf factors for stable processes

We give a series representation of the logarithm of the bivariate Laplace exponent $\kappa$ of $\alpha$-stable processes for almost all $\alpha \in (0,2]$.Comment: 16 pages. to appear in Annales IHP, 201

On exit time of stable processes

We study the exit time $\tau=\tau_{(0,\infty)}$ for 1-dimensional strictly stable processes and express its Laplace transform at $t^\alpha$ as the Laplace transform of a positive random variable with explicit density. Consequently, $\tau$ satisfies some multiplicative convolution relations. For some stable processes, e.g. for the symmetric $\frac23$-stable process, explicit formulas for the Laplace transform and the density of $\tau$ are obtained as an application

Estimates of gradient perturbation series

We give upper and lower bounds of perturbation series for transition densities, corresponding to additive gradient perturbations satisfying certain space-time integrability conditions

Estimates of the Green function for the fractional Laplacian perturbed by gradient

The Green function of the fractional Laplacian of the differential order bigger than one and the Green function of its gradient perturbations are comparable for bounded smooth multidimensional open sets if the drift function is in an appropriate Kato class.Comment: 35 pages, details added to the proof of geometrical Lemma